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Class 12-science RD SHARMA Solutions Maths Chapter 22 - Differential Equations

Differential Equations Exercise Ex. 22.1

Solution 1


Solution 2

Solution 3

Solution 4

C o n s i d e r space t h e space g i v e n space d i f f e r e n t i a l space e q u a t i o n comma space square root of 1 plus open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared end root equals open parentheses c fraction numerator d squared y over denominator d x squared end fraction close parentheses to the power of 1 third end exponent
S q u a r i n g space o n space b o t h space t h e space s i d e s comma space w e space h a v e
1 plus open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared equals open parentheses c fraction numerator d squared y over denominator d x squared end fraction close parentheses to the power of 2 over 3 end exponent
C u b i n g space o n space b o t h space t h e space s i d e s comma space w e space h a v e
open square brackets 1 plus open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared close square brackets cubed equals open curly brackets open parentheses c fraction numerator d squared y over denominator d x squared end fraction close parentheses to the power of 2 over 3 end exponent close curly brackets cubed
rightwards double arrow 1 plus open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 6 plus 3 open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared plus 3 open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 4 equals c squared open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses squared
rightwards double arrow c squared open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses squared minus open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 6 minus 3 open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 4 minus 3 open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared minus 1 equals 0
T h e space h i g h e s t space o r d e r space d i f f e r e n t i a l space c o e f f i c i e n t space i n space t h i s space
e q u a t i o n space i s space fraction numerator d squared y over denominator d x squared end fraction space a n d space i t s space p o w e r space i s space 2.
T h e r e f o r e comma space t h e space g i v e n space d i f f e r e n t i a l space e q u a t i o n space i s space a space
n o n minus l i n e a r space d i f f e r e n t i a l space e q u a t i o n space o f space s e c o n d space o r d e r space a n d space s e c o n d space d e g r e e.

Solution 5

Solution 6

C o n s i d e r space t h e space g i v e n space d i f f e r e n t i a l space e q u a t i o n comma
3 root of fraction numerator d squared y over denominator d x squared end fraction end root equals square root of fraction numerator d y over denominator d x end fraction end root
C u b i n g space o n space b o t h space t h e space s i d e s space o f space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
fraction numerator d squared y over denominator d x squared end fraction equals open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 3 over 2 end exponent
S q u a r i n g space o n space b o t h space t h e space s i d e s space o f space t h e space a b o v e space e q u a t i o n comma space w e space h a v e
open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses squared equals open square brackets open parentheses fraction numerator d y over denominator d x end fraction close parentheses to the power of 3 over 2 end exponent close square brackets squared
rightwards double arrow open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses squared equals open square brackets open parentheses fraction numerator d y over denominator d x end fraction close parentheses close square brackets cubed
rightwards double arrow open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses squared minus open square brackets open parentheses fraction numerator d y over denominator d x end fraction close parentheses close square brackets cubed equals 0
T h e space h i g h e s t space o r d e r space d i f f e r e n t i a l space c o e f f i c i e n t space i n space t h i s space e q u a t i o n space i s space fraction numerator d squared y over denominator d x squared end fraction
a n d space i t s space p o w e r space i s space 2.
T h e r e f o r e comma space t h e space g i v e n space d i f f e r e n t i a l space e q u a t i o n space i s space a space n o n minus l i n e a r space d i f f e r e n t i a l
e q u a t i o n space o f space s e c o n d space o r d e r space a n d space s e c o n d space d e g r e e.

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12


Solution 13

Solution 14

Solution 15

Solution 16


Solution 17

Solution 18


Solution 19

Solution 20

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

The order of a differential equation is the order of the highest order derivative appearing in the equation.

The degree of a differential equation is the degree of the highest order derivative.

Consider the given differential equation

In the above equation, the order of the highest order derivative is 1.

So the differential equation is of order 1.

In the above differential equation, the power of the highest order derivative is 3.

Hence, it is a differential equation of degree 3.

Since the degree of the above differential equation is 3, more than one, it is a non-linear differential equation.

Differential Equations Exercise Ex. 22.2

Solution 1

Solution 2

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8


Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

  

Solution 15(i)

C o n s i d e r space t h e space g i v e n space e q u a t i o n. comma
open parentheses 2 x plus a close parentheses squared plus y squared equals a squared.... left parenthesis 1 right parenthesis
D i f f e r e n t i a t i n g space t h e space a b o v e space e q u a t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma
2 open parentheses 2 x plus a close parentheses plus 2 y fraction numerator d y over denominator d x end fraction equals 0
rightwards double arrow open parentheses 2 x plus a close parentheses plus y fraction numerator d y over denominator d x end fraction equals 0
rightwards double arrow 2 x plus a equals minus y fraction numerator d y over denominator d x end fraction
rightwards double arrow a equals minus 2 x minus y fraction numerator d y over denominator d x end fraction
S u b s t i t u t i n g space t h e space v a l u e space o f space a space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e
open parentheses 2 x minus 2 x minus y fraction numerator d y over denominator d x end fraction close parentheses squared plus y squared equals open parentheses minus 2 x minus y fraction numerator d y over denominator d x end fraction close parentheses squared
rightwards double arrow open parentheses y fraction numerator d y over denominator d x end fraction close parentheses squared plus y squared equals open parentheses 4 x squared plus y squared open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared plus 4 x y fraction numerator d y over denominator d x end fraction close parentheses

rightwards double arrow y squared equals 4 x squared plus 4 x y fraction numerator d y over denominator d x end fraction
rightwards double arrow y squared minus 4 x squared minus 4 x y fraction numerator d y over denominator d x end fraction equals 0

Solution 15(ii)

Solution 15(iii)

C o n s i d e r space t h e space g i v e n space e q u a t i o n comma
open parentheses x minus a close parentheses squared plus 2 y squared equals a squared.... left parenthesis 1 right parenthesis
D i f f e r e n t i a t i n g space t h e space a b o v e space e q u a t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e
2 open parentheses x minus a close parentheses plus 4 y fraction numerator d y over denominator d x end fraction equals 0
rightwards double arrow open parentheses x minus a close parentheses plus 2 y fraction numerator d y over denominator d x end fraction equals 0
rightwards double arrow open parentheses x minus a close parentheses equals minus 2 y fraction numerator d y over denominator d x end fraction
rightwards double arrow a equals x plus 2 y fraction numerator d y over denominator d x end fraction
S u b s t i t u t i n g space t h e space v a l u e space o f space a space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e
open parentheses x minus x plus 2 y fraction numerator d y over denominator d x end fraction close parentheses squared plus 2 y squared equals open parentheses x plus 2 y fraction numerator d y over denominator d x end fraction close parentheses squared
rightwards double arrow 4 y squared open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared plus 2 y squared equals x squared plus 4 y squared open parentheses fraction numerator d y over denominator d x end fraction close parentheses squared plus 4 x y fraction numerator d y over denominator d x end fraction
rightwards double arrow 2 y squared minus x squared equals 4 x y fraction numerator d y over denominator d x end fraction

 

 

Solution 16(i)

Solution 16(ii)

Solution 16(iii)

Solution 16(iv)

begin mathsize 12px style straight x squared plus left parenthesis straight y minus straight b right parenthesis squared equals 1 space space space space space space space space space space space space space space space space space space space space space space space space ____ left parenthesis straight i right parenthesis
Differentiating space it space with space respect space to space straight x comma
2 straight x space plus space 2 left parenthesis straight y minus straight b right parenthesis dy over dx equals 0
straight x space plus space left parenthesis straight y minus straight b right parenthesis dy over dx equals 0
left parenthesis straight y minus straight b right parenthesis dy over dx equals negative straight x
left parenthesis straight y minus straight b right parenthesis equals fraction numerator begin display style fraction numerator negative straight x over denominator dy end fraction end style over denominator dx end fraction
Put space the space value space of space left parenthesis straight y minus straight b right parenthesis space is space equation space left parenthesis straight i right parenthesis
straight x squared open parentheses fraction numerator negative straight x over denominator begin display style dy over dx end style end fraction close parentheses squared equals 1
straight x squared open parentheses dy over dx close parentheses squared plus straight x squared equals open parentheses dy over dx close parentheses squared
straight x squared open curly brackets open parentheses dy over dx close parentheses squared plus 1 close curly brackets equals open parentheses dy over dx close parentheses squared
end style

Solution 16(v)

Solution 16(vi)

Solution 16(vii)

Solution 16(viii)

Solution 16(ix)

Solution 16(x)

Solution 17

Solution 18

Solution 19

Differential Equations Exercise Ex. 22.3

Solution 1

Solution 2

Solution 3

begin mathsize 12px style straight y equals ae to the power of 2 straight x end exponent plus be to the power of negative straight x end exponent space space space space space space space space space space space space space space space space space space space space space ___ left parenthesis straight i right parenthesis
Differentiating space it space with space respect space to space straight x comma
dy over dx equals 2 ae to the power of 2 straight x end exponent minus be to the power of negative straight x end exponent space space space space space space space space space space space space space ___ left parenthesis ii right parenthesis
Differentiating space it space with space respect space to space straight x comma
fraction numerator straight d squared straight y over denominator dx squared end fraction equals 4 ae to the power of 2 straight x end exponent minus be to the power of negative straight x end exponent space space space space space space space space space space space space space ___ left parenthesis iii right parenthesis
Now comma
fraction numerator straight d squared straight y over denominator dx squared end fraction minus dy over dx minus 2 straight y
equals left parenthesis 4 ae to the power of 2 straight x end exponent plus be to the power of negative straight x end exponent right parenthesis minus left parenthesis 2 ae to the power of 2 straight x end exponent minus be to the power of negative straight x end exponent right parenthesis minus 2 left parenthesis ae to the power of 2 straight x end exponent plus be to the power of negative straight x end exponent right parenthesis
equals 4 ae to the power of 2 straight x end exponent plus be to the power of negative straight x end exponent minus 2 ae to the power of 2 straight x end exponent plus be to the power of negative straight x end exponent minus 2 ae to the power of 2 straight x end exponent minus 2 be to the power of negative straight x end exponent
equals 4 ae to the power of 2 straight x end exponent minus 4 ae to the power of 2 straight x end exponent plus 2 be to the power of negative straight x end exponent minus 2 be to the power of negative straight x end exponent
equals 0
S o comma
fraction numerator d squared y over denominator d x squared end fraction minus fraction numerator d y over denominator d x end fraction minus 2 y equals 0 end style

Solution 4

Solution 5

Solution 6

Solution 7

begin mathsize 12px style straight y equals straight a over straight x plus straight b space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space ___ left parenthesis straight i right parenthesis
Differentiating space it space with space respect space to space straight x comma
dy over dx equals negative straight a over straight x squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space ___ left parenthesis ii right parenthesis
Differentiating space it space with space respect space to space straight x comma
fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator 2 straight a over denominator straight x cubed end fraction
equals negative 2 over straight x open parentheses negative straight a over straight x squared close parentheses
fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 2 over straight x open parentheses dy over dx close parentheses
fraction numerator straight d squared straight y over denominator dx squared end fraction plus 2 over straight x open parentheses dy over dx close parentheses equals 0 end style

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

  

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

left parenthesis 1 minus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx minus straight m squared straight y equals 0
straight y equals straight e to the power of mcos to the power of negative 1 end exponent straight x end exponent
dy over dx equals fraction numerator me to the power of mcos to the power of negative 1 end exponent straight x end exponent over denominator negative square root of 1 minus straight x squared end root end fraction
dy over dx equals fraction numerator negative my over denominator square root of 1 minus straight x squared end root end fraction........ left parenthesis straight i right parenthesis space
fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator square root of left parenthesis 1 minus straight x squared right parenthesis end root. open parentheses negative straight m begin display style dy over dx end style close parentheses minus left parenthesis negative my right parenthesis begin display style fraction numerator left parenthesis negative 2 straight x right parenthesis over denominator 2 square root of left parenthesis 1 minus straight x squared right parenthesis end root end fraction end style over denominator left parenthesis 1 minus straight x squared right parenthesis end fraction space left square bracket From space left parenthesis straight i right parenthesis right square bracket
fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator left parenthesis negative straight m right parenthesis open parentheses begin display style negative my end style close parentheses minus straight x begin display style dy over dx end style over denominator left parenthesis 1 minus straight x squared right parenthesis end fraction left square bracket From space left parenthesis straight i right parenthesis right square bracket
left parenthesis 1 minus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight m squared straight y minus straight x dy over dx
left parenthesis 1 minus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx minus straight m squared straight y equals 0
Hence space Proved

Solution 18

Solution 19

Solution 20

begin mathsize 12px style straight y equals straight e to the power of negative straight x end exponent plus ax plus straight b
Differentiating space it space with space respect space to space straight x comma
dy over dx equals negative straight e to the power of negative straight x end exponent plus straight a
Differentiating space it space with space respect space to space straight x comma
fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight e to the power of negative straight x end exponent
1 over straight e to the power of negative straight x end exponent fraction numerator straight d squared straight y over denominator dx squared end fraction equals 1
straight e to the power of straight x fraction numerator straight d squared straight y over denominator dx squared end fraction equals 1 end style

Solution 21(i)

begin mathsize 12px style straight y equals ax space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis
Differentiating space it space with space respect space to space straight x comma
dy over dx equals straight a
equals ax over straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because straight x element of straight R minus left curly bracket 0 right curly bracket right square bracket
dy over dx equals straight y over straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket Using space equation space left parenthesis straight i right parenthesis right square bracket
straight x dy over dx equals straight y
So comma space straight y space equals space ax space is space the space solution space of space the space given space equation. end style

Solution 21(ii)

Solution 21(iii)

  

Solution 21(iv)

Solution 21(v)

Differential Equations Exercise Ex. 22.4

Solution 1

Solution 2

Solution 3

begin mathsize 12px style Here comma space straight y space equals space sin space straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight i right parenthesis
Differentiating space it space with space respect space to space straight x comma
dy over dx equals cos space straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis ii right parenthesis
Again space differentiating space it space with space respect space to space straight x comma
fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative sin space straight x space
fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative straight y
fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y equals 0
So comma space straight y equals sin space straight x space is space straight a space solution space of space the space equation.
Put space space space space space straight x equals 0 space in space equation space left parenthesis straight i right parenthesis comma
rightwards double arrow space space space space space straight y space equals space sin space 0
rightwards double arrow space space space space space straight y equals 0
rightwards double arrow space space space space space straight y left parenthesis 0 right parenthesis space equals 0
Put space straight x space equals 0 space in space equation space left parenthesis ii right parenthesis
straight y apostrophe space equals cos space 0
straight y apostrophe space equals 1
rightwards double arrow straight y apostrophe left parenthesis 0 right parenthesis space equals space 1 end style

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Differential Equations Exercise Ex. 22.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

w h e r e space x not equal to open parentheses 2 n plus 1 close parentheses pi comma space n element of Z

Solution 6

Solution 7

w h e r e space x element of R

Solution 8

fraction numerator d y over denominator d x end fraction equals log x
rightwards double arrow d y equals log x cross times d x
rightwards double arrow integral d y equals integral log x d x
rightwards double arrow y equals log x cross times integral 1 d x minus integral open parentheses 1 over x integral 1 d x close parentheses d x plus C space space space space space open square brackets U sin g space i n t e g r a t i o n space b y space p a r t s close square brackets
rightwards double arrow y equals x log x minus integral d x plus C
rightwards double arrow y equals x log x minus x plus C
rightwards double arrow y equals x open parentheses log x minus 1 close parentheses plus C comma space w h e r e space x element of open parentheses 0 comma infinity close parentheses

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

begin mathsize 12px style cos space straight x space dy over dx minus cos space 2 space straight x space equals space cos space 3 space straight x
cos space straight x space dy over dx minus cos space 3 space straight x space equals space cos space 2 space straight x
dy over dx equals fraction numerator 4 cos cubed space straight x minus 3 cos space straight x space plus 2 space cos squared straight x space minus 1 over denominator cos space straight x end fraction
dy over dx equals fraction numerator 4 cos cubed straight x over denominator cos space straight x end fraction minus fraction numerator 3 cos space straight x over denominator cos space straight x end fraction plus fraction numerator 2 cos squared straight x over denominator cos space straight x end fraction minus fraction numerator 1 over denominator cos space straight x end fraction
dy over dx equals 4 cos squared straight x minus 3 plus 2 space cos space straight x minus space sec space straight x
dy over dx equals 4 open parentheses fraction numerator cos space 2 straight x plus 1 over denominator 2 end fraction close parentheses minus 32 space cos space straight x space minus space sec space straight x
dy equals left parenthesis 2 space cos space 2 space straight x space plus space 2 minus 3 plus 2 space cos space straight x space minus sec space straight x right parenthesis dx
integral dy equals integral left parenthesis 2 cos space 2 straight x space minus 1 plus 2 space cos space straight x minus space sec space straight x right parenthesis dx
straight y space equals space sin space 2 straight x space minus straight x plus 2 space sin space straight x space minus log open vertical bar sec space straight x space plus space tan space straight x close vertical bar plus straight c end style

Solution 16

Solution 17

Solution 18

begin mathsize 12px style left parenthesis 1 plus straight x squared right parenthesis dy over dx minus straight x equals 2 tan to the power of negative 1 end exponent straight x
left parenthesis 1 plus straight x squared right parenthesis dy over dx equals 2 tan to the power of negative 1 end exponent straight x plus straight x
dy equals open parentheses fraction numerator 2 tan to the power of negative 1 end exponent straight x plus straight x over denominator 1 plus straight x squared end fraction close parentheses dx
integral dy equals integral open parentheses fraction numerator 2 tan to the power of negative 1 end exponent straight x plus straight x over denominator 1 plus straight x squared end fraction close parentheses dx
straight y equals integral left parenthesis 2 straight t plus tant right parenthesis dt space space space space space space space space space space space space space space space space left square bracket tan to the power of negative 1 end exponent straight x equals straight t right square bracket
equals 1 half log open vertical bar 1 plus straight x squared close vertical bar plus left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared plus straight c end style

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25


Solution 26

begin mathsize 12px style straight x left parenthesis straight x squared minus 1 right parenthesis dy over dx equals 1 comma straight y left parenthesis 2 right parenthesis equals 0
dy over dx equals fraction numerator 1 over denominator straight x left parenthesis straight x squared minus 1 right parenthesis end fraction
dy equals fraction numerator 1 over denominator straight x left parenthesis straight x squared minus 1 right parenthesis end fraction dx
integral dy equals integral open parentheses fraction numerator 1 over denominator straight x open parentheses straight x squared minus 1 close parentheses end fraction close parentheses dx
straight y equals 1 half integral fraction numerator 1 over denominator straight x minus 1 end fraction dx minus integral 1 over straight x dx plus 1 half integral fraction numerator 1 over denominator straight x plus 1 end fraction dx
equals 1 half log open vertical bar straight x minus 1 close vertical bar minus log open vertical bar straight x close vertical bar plus 1 half log open vertical bar straight x plus 1 close vertical bar plus straight c
Putting space straight x equals 2 comma space straight y equals 0 comma space we space have
straight y equals 1 half log open vertical bar straight x minus 1 close vertical bar minus log open vertical bar straight x close vertical bar plus 1 half log open vertical bar straight x plus 1 close vertical bar plus straight c space
0 equals 1 half log open vertical bar 2 minus 1 close vertical bar minus log open vertical bar 2 close vertical bar plus 1 half log open vertical bar 2 plus 1 close vertical bar plus straight c space
straight c equals log open vertical bar 2 close vertical bar minus 1 half log open vertical bar 3 close vertical bar
Putting space the space value space ofc comma space we space have
straight y equals 1 half log open vertical bar straight x minus 1 close vertical bar minus log open vertical bar straight x close vertical bar plus 1 half log open vertical bar straight x plus 1 close vertical bar plus straight c
equals log 4 over 3 open parentheses fraction numerator straight x squared minus 1 over denominator straight x squared end fraction close parentheses end style

Differential Equations Exercise Ex. 22.6

Solution 1

Solution 2

Solution 3

Solution 4

Differential Equations Exercise Ex. 22.7

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

begin mathsize 12px style cos space straight x space cos space straight y space dy over dx equals negative sin space straight x space sin space straight y
fraction numerator cos space straight y over denominator sin space straight y end fraction dy space equals space minus fraction numerator sin space straight x over denominator cos space straight x end fraction dx
integral cot space ydy space equals negative integral tan space xdx
log space sin space straight y space equals space log space cos space straight x space plus space log space straight c
sin space straight y space equals space straight c space cos space straight x end style

Solution 26

Solution 27

Solution 28

Solution 29

begin mathsize 12px style left parenthesis straight y space plus space xy right parenthesis dx space plus open parentheses straight x minus xy squared close parentheses dy space equals space 0
straight y left parenthesis 1 plus straight x right parenthesis dx equals open parentheses xy squared minus straight x close parentheses dy
straight y left parenthesis 1 plus straight x right parenthesis dx equals straight x open parentheses straight y squared minus 1 close parentheses dy
fraction numerator open parentheses straight y squared minus 1 close parentheses dy over denominator straight y end fraction equals fraction numerator 1 plus straight x over denominator straight x end fraction dx
integral open parentheses straight y minus 1 over straight y close parentheses dy equals integral open parentheses 1 over straight x plus 1 close parentheses dx
straight y squared over 2 minus log open vertical bar straight y close vertical bar equals log open vertical bar straight x close vertical bar plus straight x plus straight c subscript 1
straight y squared over 2 minus straight x minus log open vertical bar straight y close vertical bar minus log open vertical bar straight x close vertical bar equals straight c subscript 1
log open vertical bar straight x close vertical bar plus straight x plus log open vertical bar straight y close vertical bar minus straight y squared over 2 equals straight c end style

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

begin mathsize 12px style dy over dx equals straight e to the power of straight x plus straight y end exponent plus straight e to the power of negative straight x plus straight y end exponent
equals straight e to the power of straight x space xe to the power of straight y plus straight e to the power of negative straight x end exponent space xe to the power of straight y
dy over dx equals straight e to the power of straight y open parentheses straight e to the power of straight x plus straight e to the power of negative straight x end exponent close parentheses
dy over straight e to the power of straight y equals open parentheses straight e to the power of straight x plus straight e to the power of negative straight x end exponent close parentheses dx
integral straight e to the power of negative straight y end exponent dy equals integral open parentheses straight e to the power of straight x plus straight e to the power of negative straight x end exponent close parentheses dx
minus straight e to the power of negative straight y end exponent equals straight e to the power of straight x minus straight e to the power of negative straight x end exponent plus straight c
straight e to the power of negative straight x end exponent minus straight e to the power of negative straight y end exponent equals straight e to the power of straight x plus straight c end style

Solution 36

begin mathsize 12px style dy over dx equals open parentheses cos squared straight x minus sin squared straight x close parentheses cos squared straight y
fraction numerator dy over denominator cos squared straight y end fraction equals open parentheses cos squared straight x minus sin squared straight x close parentheses dx
integral sec squared ydy equals integral cos 2 xdx
tan space straight y equals fraction numerator sin 2 straight x over denominator 2 end fraction plus straight c end style

Solution 37(i)

Solution 37(ii)

Solution 38(i)

Solution 38(ii)

Solution 38(iii)

  

Solution 38(iv)

  

Solution 39

Solution 40

Solution 41

Solution 42

begin mathsize 12px style dy over dx equals 2 straight e to the power of straight x straight y cubed comma straight y left parenthesis 0 right parenthesis equals 1 half
integral dy over straight y cubed equals integral 2 straight e to the power of straight x dx
minus fraction numerator 1 over denominator 2 straight y squared end fraction equals 2 straight e to the power of straight x plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
Put space straight x space equals space 0 comma space straight y space equals space 1 half
minus 4 over 2 equals 2 straight e to the power of 0 plus straight c
minus 2 equals 2 plus straight c
straight c equals negative 4
Put space straight c equals negative 4 space in space equation space left parenthesis straight i right parenthesis
minus fraction numerator 1 over denominator 2 straight y squared end fraction equals 2 straight e to the power of straight x minus 4
minus 1 equals 4 straight e to the power of straight x straight y squared minus 8 straight y squared
minus 1 equals negative straight y squared left parenthesis 8 minus 4 straight e to the power of straight x right parenthesis
straight y squared left parenthesis 8 minus 4 straight e to the power of straight x right parenthesis equals 1 end style

Solution 43

begin mathsize 12px style dr over dt equals negative rt comma space straight r left parenthesis 0 right parenthesis space equals space straight r subscript 0
integral dr over straight r equals negative integral tdt
log open vertical bar straight r close vertical bar equals negative straight t squared over 2 plus straight c space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative negative left parenthesis straight i right parenthesis
Put space straight t space equals space 0 comma space straight r space equals space straight r subscript 0 space inequation space left parenthesis straight i right parenthesis comma
log open vertical bar straight r subscript 0 close vertical bar equals 0 plus straight c
log open vertical bar straight r subscript 0 close vertical bar equals straight c
Now comma
log open vertical bar straight r close vertical bar equals negative straight t squared over 2 plus log open vertical bar straight r subscript 0 close vertical bar
straight r over straight r subscript 0 equals straight e to the power of negative straight t squared over 2 end exponent
straight r equals straight r subscript 0 straight e to the power of negative straight t squared over 2 end exponent end style

Solution 44

Solution 45(i)

Solution 45(ii)

Solution 45(iii)

begin mathsize 12px style dy over dx equals 2 straight e to the power of 2 straight x end exponent straight y squared comma space straight y space left parenthesis 0 right parenthesis equals negative 1
integral dy over straight y squared equals integral 2 straight e to the power of 2 straight x end exponent dx
minus 1 over straight y equals fraction numerator 2 straight e to the power of 2 straight x end exponent over denominator 2 end fraction plus straight c
minus 1 over straight y equals straight e to the power of 2 straight x end exponent plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
Put space straight y space equals space minus 1 comma space straight x space equals space 0
1 equals straight e to the power of 0 plus straight c
1 equals 1 plus straight c
straight c equals 0
Put space straight c equals 0 space in space equation space left parenthesis straight i right parenthesis comma
minus 1 over straight y equals straight e to the power of 2 straight x end exponent
straight y equals negative straight e to the power of negative 2 straight x end exponent end style

Solution 45(iv)

begin mathsize 12px style cos space straight y dy over dx equals straight e to the power of straight x comma space straight y space left parenthesis 0 right parenthesis equals straight pi over 2
integral cos space ydy equals integral straight e to the power of straight x dx
sin space straight y equals straight e to the power of straight x plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
Put space straight x space equals 0 comma space straight y equals straight pi over 2
sin open parentheses straight pi over 2 close parentheses equals straight e to the power of 0 plus straight c
1 equals 1 plus straight c
straight c equals 0
Put space straight c equals 0 space in space equation space left parenthesis straight i right parenthesis comma
sin space straight y space equals space straight e to the power of straight x
straight y equals sin to the power of negative 1 end exponent open parentheses straight e to the power of straight x close parentheses end style

Solution 45(v)

Solution 45(vi)

begin mathsize 12px style dy over dx equals 1 plus straight x squared plus straight y squared plus straight x squared straight y squared comma space straight y left parenthesis 0 right parenthesis equals 1
equals left parenthesis 1 plus straight x squared right parenthesis left parenthesis 1 plus straight y squared right parenthesis
integral fraction numerator dy over denominator 1 plus straight y squared end fraction equals integral left parenthesis 1 plus straight x squared right parenthesis dx
tan to the power of negative 1 end exponent straight y equals straight x plus straight x cubed over 3 plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
Put space space straight x equals 0 comma space straight y equals 1
tan to the power of negative 1 end exponent straight y equals straight x plus straight x cubed over 3 plus straight c
straight c equals straight pi over 4
Put space straight c equals straight pi over 4 space in space equation space left parenthesis straight i right parenthesis
tan to the power of negative 1 end exponent straight y equals straight x plus straight x cubed over 3 plus straight pi over 4 end style

Solution 45(vii)

begin mathsize 12px style xy dy over dx equals open parentheses straight x plus 2 close parentheses open parentheses straight y plus 2 close parentheses comma space straight y open parentheses 1 close parentheses equals negative 1
fraction numerator ydy over denominator open parentheses straight y plus 2 close parentheses end fraction equals fraction numerator open parentheses straight x plus 2 close parentheses over denominator straight x end fraction dx
integral open parentheses 1 minus fraction numerator 2 over denominator straight y plus 2 end fraction close parentheses dy equals integral open parentheses 1 plus 2 over straight x close parentheses dx
straight y minus straight x minus 2 log left parenthesis straight y plus 2 right parenthesis minus 2 logx equals straight c
Put space straight x equals 1 comma space straight y equals negative 1
minus 1 minus 1 minus 2 log left parenthesis negative 1 plus 2 right parenthesis minus 2 log 1 equals straight c
rightwards double arrow negative 2 equals straight c
Thus comma space we space have
straight y minus straight x minus 2 log left parenthesis straight y plus 2 right parenthesis minus 2 logx equals negative 2
end style

Solution 45(viii)

  

Solution 45(ix)

  

Solution 46

Solution 47

Solution 48

Solution 49

begin mathsize 12px style dy over straight e to the power of dx equals straight x plus 1
dy over dx equals log left parenthesis straight x plus 1 right parenthesis comma space straight y equals 3 space at space straight x equals 0
integral dy equals integral log left parenthesis straight x plus 1 right parenthesis dx
straight y equals log open vertical bar straight x plus 1 close vertical bar straight x integral 1 cross times dx minus integral open parentheses fraction numerator 1 over denominator straight x plus 1 end fraction cross times integral 1 dx close parentheses dx plus straight c
Using space in space tegration space by space parts
straight y equals straight x space log open vertical bar straight x space plus 1 close vertical bar minus integral fraction numerator straight x over denominator straight x plus 1 end fraction dx plus straight c
straight y equals straight x space log open vertical bar straight x plus 1 close vertical bar minus open parentheses integral open parentheses 1 minus fraction numerator 1 over denominator straight x plus 1 end fraction close parentheses dx close parentheses plus straight c
equals space straight x space log space open vertical bar straight x plus 1 close vertical bar minus open parentheses straight x minus log open vertical bar straight x plus 1 close vertical bar close parentheses plus straight c
straight y equals xlog open vertical bar straight x plus 1 close vertical bar minus straight x plus log open vertical bar straight x plus 1 close vertical bar plus straight c
straight y equals open parentheses straight x plus 1 close parentheses log open vertical bar straight x plus 1 close vertical bar minus straight x plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
Put space straight y space equals 3 space and space straight x space equals 0
3 equals 0 minus 0 plus straight c
straight c equals 3
Put space straight c space equals space 3 space in space equation space left parenthesis straight i right parenthesis comma
straight y equals left parenthesis straight x plus 1 right parenthesis log open vertical bar straight x plus 1 close vertical bar minus straight x plus 3 end style

Solution 50

begin mathsize 12px style cos space ydy space plus space cos space straight x space sin space ydx equals 0
cos space ydy equals negative cos space straight x space sin space ydx
fraction numerator cos space straight y over denominator sin space straight y end fraction dy equals negative cos space xdx
integral cot space ydy equals negative integral cos space xdx
log open vertical bar sin space straight y close vertical bar equals negative sin space straight x space plus straight c space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
put space straight y space equals space straight pi over 2 and space straight x space equals straight pi over 2
log space open vertical bar sin straight pi over 2 close vertical bar equals negative sin straight pi over 2 plus straight c
0 equals negative 1 plus straight c
straight c equals 1
Put space straight c space equals 1 space in space equation space left parenthesis 1 right parenthesis comma
log open vertical bar sin space straight y close vertical bar equals 1 minus sin space straight x
log open vertical bar sin space straight y close vertical bar plus sin space straight x equals 1 end style

Solution 51

begin mathsize 12px style dy over dx equals negative 4 xy squared comma space straight y equals 1 space when space straight x space equals 0
integral dy over straight y squared equals negative 4 integral xdx
minus 1 over straight y equals negative 4 straight x squared over 2 plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space minus negative negative negative left parenthesis straight i right parenthesis
Put space straight y space equals 1 space and space straight x space equals 0
minus 1 equals 0 plus straight c
straight c equals negative 1
Put space straight c equals negative 1 space in space equation space left parenthesis ii right parenthesis comma
minus 1 over straight y equals negative 2 straight x squared minus 1
1 over straight y equals 2 straight x squared plus 1
straight y equals fraction numerator 1 over denominator 2 straight x squared plus 1 end fraction end style

Solution 52

Solution 53

Solution 54

begin mathsize 12px style Let space the space rate space of space change space of space volume space of space the space balloon space be space straight k left parenthesis where space straight k space is space straight a space constant right parenthesis
rightwards double arrow dv over dt equals straight k
rightwards double arrow straight d over dt open parentheses 4 over 3 πr cubed close parentheses equals straight k space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets Volume space of space sphere equals 4 over 3 πr cubed close square brackets
rightwards double arrow 4 over 3 straight pi times 3 straight r squared times dr over dt equals straight k
rightwards double arrow 4 πr squared dr equals straight k space dt
Intrgrating space both space sides comma space we space get colon
4 straight pi integral straight r squared dr equals straight k integral dt
rightwards double arrow 4 straight pi times straight r squared over 3 equals kt plus straight c
rightwards double arrow 4 πr cubed equals 3 open parentheses kt plus straight c close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space........ left parenthesis straight i right parenthesis
Now comma space at space straight t space equals space 0 comma space straight r space equals space 3 colon
4 straight pi cross times 3 cubed equals 3 left parenthesis straight k cross times 0 plus straight c right parenthesis
108 straight pi equals 3 straight c
straight c equals 36 straight pi end style

begin mathsize 12px style At space straight t space equals 3 comma space straight r equals 6
4 straight pi cross times 6 cubed equals 3 left parenthesis straight k cross times 3 plus straight c right parenthesis
864 straight pi equals 3 left parenthesis 3 straight k plus 36 straight pi right parenthesis
3 straight k equals negative 288 straight pi minus 36 straight pi equals 252 straight pi
straight k equals 84 straight pi
Substituting space the space values space of space straight k space and space straight C space in space equation space left parenthesis 1 right parenthesis comma space we space get colon
4 πr cubed equals 3 open square brackets 84 πt plus 36 straight pi close square brackets
rightwards double arrow 4 πr cubed equals 4 straight pi left parenthesis 63 straight t plus 27 right parenthesis
rightwards double arrow straight r cubed equals 63 straight t plus 27
rightwards double arrow straight r equals left parenthesis 63 straight t plus 27 right parenthesis to the power of 1 third end exponent space Thus comma space the space radius space of space the space balloon space after space straight t space seconds space is space left parenthesis 63 straight t plus 27 right parenthesis to the power of 1 third end exponent. end style

 

Solution 55

Let p, t and represent the principal, time, and rate of interest respectively.

It is given that the principal increases continuously at the rate of r% per year.

begin mathsize 12px style rightwards double arrow dp over dt equals open parentheses straight r over 100 close parentheses straight p
rightwards double arrow dp over straight p equals open parentheses straight r over 100 close parentheses dt end style

Integrating both side, we get:

begin mathsize 12px style integral dp over straight p equals straight r over 100 integral dt
rightwards double arrow log space straight p space equals space rt over 100 plus straight k
rightwards double arrow straight p equals straight e to the power of rt over 100 plus straight k end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space...... left parenthesis straight i right parenthesis
It space is space given space that space when space straight t space equals 0 comma space straight p space 100.
rightwards double arrow 100 space equals straight e to the power of straight k space space space space space space space space space space space space space space space space space space space space space..... left parenthesis 2 right parenthesis
Now comma space it space calligraphic l equals 10 comma space then space straight p space equals 2 space cross times 100 space equals 200.
200 equals straight e to the power of calligraphic l over 10 plus straight k end exponent
rightwards double arrow 200 space equals straight e to the power of calligraphic l over 10 end exponent. straight e to the power of straight k
rightwards double arrow 200 space equals straight e to the power of calligraphic l over 10 end exponent.100 space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis From left parenthesis 2 right parenthesis right parenthesis
rightwards double arrow straight e to the power of calligraphic l over 10 end exponent equals 2
rightwards double arrow straight r over 10 log subscript straight e 2
rightwards double arrow straight r over 10 equals 0.6931
rightwards double arrow straight r space equals 6.931
Hence comma space the space value space of space straight r space is space 6.93 percent sign end style

 

Solution 56

Solution 57

..

Solution 58

Differential Equations Exercise Ex. 22.8

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

2 v minus v fraction numerator d v over denominator d x end fraction equals fraction numerator d v over denominator d x end fraction
rightwards double arrow 2 v equals v fraction numerator d v over denominator d x end fraction plus fraction numerator d v over denominator d x end fraction
rightwards double arrow 2 v equals open parentheses v plus 1 close parentheses fraction numerator d v over denominator d x end fraction
rightwards double arrow fraction numerator open parentheses v plus 1 close parentheses over denominator v end fraction d v equals 2 d x

Solution 10

Solution 11

  

Differential Equations Exercise Ex. 22.9

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

begin mathsize 12px style dy over dx equals fraction numerator straight x plus straight y over denominator straight x minus straight y end fraction
Here space it space is space straight a space homogeneous space equation
put space straight y equals vx
And
dy over dx equals straight v plus straight x dv over dx
So comma
straight v plus straight x dv over dx equals fraction numerator 1 plus straight v over denominator 1 minus straight v end fraction
straight x dv over dx equals fraction numerator 1 plus straight v over denominator 1 minus straight v end fraction minus straight v
straight x dv over dx equals fraction numerator 1 plus straight v squared over denominator 1 minus straight v end fraction
fraction numerator 1 minus straight v over denominator 1 plus straight v squared end fraction dv equals dx over straight x
integral fraction numerator 1 minus straight v over denominator 1 plus straight v squared end fraction dv equals integral dx over straight x
integral fraction numerator 1 over denominator 1 plus straight v squared end fraction dv minus 1 half integral fraction numerator 2 straight v over denominator 1 plus straight v squared end fraction dv equals integral dx over straight x
tan to the power of negative 1 end exponent straight v minus 1 half log left parenthesis 1 plus straight v squared right parenthesis equals log space straight x space plus straight c
tan to the power of negative 1 end exponent straight y over straight x equals 1 half log open parentheses straight x squared plus straight y squared close parentheses plus straight c end style

Solution 7

Solution 8

C o n s i d e r space t h e space g i v e n space d i f f e r e n t i a l space e q u a t i o n
space x squared fraction numerator d y over denominator d x end fraction equals x squared minus 2 y squared plus x y
rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator x squared minus 2 y squared plus x y over denominator x squared end fraction
T h i s space i s space a space h o m o g e n e o u s space d i f f e r e n t i a l space e q u a t i o n.
S u b s t i t u t i n g space y equals v x space a n d space fraction numerator d y over denominator d x end fraction equals v plus x fraction numerator d v over denominator d x end fraction comma space w e space h a v e
v plus x fraction numerator d v over denominator d x end fraction equals fraction numerator x squared minus 2 v squared cross times x squared plus x cross times v cross times x over denominator x squared end fraction
rightwards double arrow v plus x fraction numerator d v over denominator d x end fraction equals 1 minus 2 v squared plus v
rightwards double arrow x fraction numerator d v over denominator d x end fraction equals 1 minus 2 v squared
rightwards double arrow fraction numerator d v over denominator 1 minus 2 v squared end fraction equals fraction numerator d x over denominator x end fraction
rightwards double arrow fraction numerator d v over denominator v squared minus begin display style 1 half end style end fraction equals minus 2 fraction numerator d x over denominator x end fraction
rightwards double arrow integral fraction numerator d v over denominator open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared minus v squared end fraction equals 2 integral fraction numerator d x over denominator x end fraction
rightwards double arrow integral fraction numerator d v over denominator open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared minus v squared end fraction equals 2 integral fraction numerator d x over denominator x end fraction
rightwards double arrow fraction numerator square root of 2 over denominator 2 end fraction log open parentheses fraction numerator fraction numerator 1 over denominator square root of 2 end fraction plus v over denominator fraction numerator 1 over denominator square root of 2 end fraction minus v end fraction close parentheses equals 2 log x plus log C
rightwards double arrow fraction numerator 1 over denominator square root of 2 end fraction log open parentheses fraction numerator fraction numerator 1 over denominator square root of 2 end fraction plus begin display style y over x end style over denominator fraction numerator 1 over denominator square root of 2 end fraction minus y over x end fraction close parentheses equals 2 log x plus log C
rightwards double arrow fraction numerator 1 over denominator square root of 2 end fraction log open parentheses fraction numerator x plus y square root of 2 over denominator x minus y square root of 2 end fraction close parentheses equals 2 log x plus log C
rightwards double arrow fraction numerator 1 over denominator square root of 2 end fraction log open parentheses fraction numerator x plus y square root of 2 over denominator x minus y square root of 2 end fraction close parentheses equals log x squared plus log C
rightwards double arrow log open parentheses fraction numerator x plus y square root of 2 over denominator x minus y square root of 2 end fraction close parentheses to the power of fraction numerator 1 over denominator square root of 2 end fraction end exponent equals log C x squared
rightwards double arrow open parentheses fraction numerator x plus y square root of 2 over denominator x minus y square root of 2 end fraction close parentheses to the power of fraction numerator 1 over denominator square root of 2 end fraction end exponent equals C x squared
rightwards double arrow open parentheses fraction numerator x plus y square root of 2 over denominator x minus y square root of 2 end fraction close parentheses equals open parentheses C x squared close parentheses to the power of square root of 2 end exponent

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13


Solution 14

Solution 15


Solution 16

Solution 17

Solution 18

Solution 19

begin mathsize 12px style dy over dx equals straight y over straight x plus sin open parentheses straight y over straight x close parentheses
Here space it space is space straight a space homogeneous space equation
Put space space space space space space space space space straight y space equals space vx
And
dy over dx equals straight v plus straight x dv over dx
So comma
straight v plus straight x dv over dx equals straight v plus sin space straight v
straight x dv over dx equals sin space straight v
cosecvdv space equals dx over straight x
integral cosecvdv equals integral dx over straight x
log space tan space straight v over 2 equals log space straight x plus log space straight c
tan straight v over 2 equals Cx
tan fraction numerator straight y over denominator 2 straight x end fraction equals Cx end style

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24


Solution 25

begin mathsize 12px style open parentheses 1 plus straight e to the power of straight x over straight y end exponent close parentheses dx plus straight e to the power of straight x over straight y end exponent open parentheses 1 minus straight x over straight y close parentheses dy equals 0
Here space it space is space straight a space homogeneous space equation
Put space space space space space space space space space space space space space space straight x equals vy
And
dx over dy equals straight v plus straight y dv over dy
So comma
dx over dy equals negative fraction numerator straight e to the power of begin display style straight x over straight y end style end exponent open parentheses 1 minus begin display style straight x over straight y end style close parentheses over denominator open parentheses 1 plus straight e to the power of begin display style straight x over straight y end style end exponent close parentheses end fraction
straight v plus straight y dv over dy equals negative fraction numerator straight e to the power of begin display style vy over straight y end style end exponent open parentheses 1 minus begin display style vy over straight y end style close parentheses over denominator blank end fraction
equals negative straight e to the power of straight v fraction numerator left parenthesis 1 minus straight v right parenthesis over denominator left parenthesis 1 plus straight e to the power of straight v right parenthesis end fraction
straight y dv over dy equals negative fraction numerator straight e to the power of straight v left parenthesis 1 minus straight v right parenthesis over denominator open parentheses 1 plus straight e to the power of straight v close parentheses end fraction
equals fraction numerator negative straight e to the power of straight v left parenthesis 1 minus straight v right parenthesis minus straight v left parenthesis 1 plus straight e to the power of straight v right parenthesis over denominator left parenthesis 1 plus straight e to the power of straight v right parenthesis end fraction
fraction numerator left parenthesis 1 plus straight e to the power of straight v right parenthesis over denominator negative straight e to the power of straight v open parentheses 1 minus straight v close parentheses minus straight v open parentheses 1 plus straight e to the power of straight v close parentheses end fraction dv equals dy over straight y
straight x plus ye to the power of straight x divided by straight y end exponent equals straight c end style

Solution 26



Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33


Solution 34

Solution 35

Solution 36(i)

Solution 36(ii)

Solution 36(iii)

begin mathsize 12px style dy over dx minus straight y over straight x plus cosec straight y over straight x equals 0 comma space straight y left parenthesis 1 right parenthesis equals 0
Here space it space is space straight a space homogeneous space equation
Put space space space space space space space space space straight y space equals vx
And
dy over dx equals straight v plus straight x dv over dx
So comma
straight v plus straight x dv over dx equals vx over straight x minus cosec vx over straight x
straight x dv over dx equals straight v minus cosecv minus straight v
equals negative cosecv
dv over cosecv equals negative dx over straight x
sin space vdv equals negative dx over straight x
minus cos space straight v equals negative log open vertical bar straight x close vertical bar plus straight c
minus cos straight y over straight x equals negative log open vertical bar straight x close vertical bar plus straight c
Now space putting space straight y equals 0 comma space straight x equals 1 comma space we space have
straight c equals negative 1
Now
minus cos straight y over straight x plus 1 equals negative log open vertical bar straight x close vertical bar
log open vertical bar straight x close vertical bar equals cos straight y over straight x minus 1 end style

Solution 36(iv)

Solution 36(v)


Solution 36(vi)


Solution 36(vii)

Solution 36(viii)

  

Solution 36(ix)

Error converting from MathML to accessible text.

Solution 37

Solution 38

Solution 39

  

Differential Equations Exercise Ex. 22.10

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

begin mathsize 12px style Here comma space open parentheses sin space straight x close parentheses dy over dx plus space straight y space cos space straight x space equals 2 space sin squared straight x space cos space straight x
dy over dx plus straight y space cot space straight x space equals space 2 space sin space straight x space cos space straight x
It space is space straight a space linear space differential space equation. space Comparing space it space with comma
dy over dx plus py equals straight Q
straight p equals cot space straight x comma space straight Q space equals 2 sin space straight x space cos space straight x
straight I. straight F. space space equals straight e to the power of integral pdx end exponent
space space space space space space space space space equals straight e to the power of integral cotxdx end exponent
space space space space space space space space space equals straight e to the power of log space sin space straight x end exponent
space space space space space space space space space equals space sin space straight x
solution space of space the space equation space is space given space by comma
straight y space cross times open parentheses straight I. straight F close parentheses equals integral straight Q cross times left parenthesis straight I. straight F right parenthesis dx plus straight c
straight y left parenthesis sin space straight x right parenthesis equals integral 2 sin space straight x space cos space straight x left parenthesis sin space straight x right parenthesis space dx space plus straight c
ysin space straight x equals space left parenthesis 2 divided by 3 right parenthesis sin to the power of logical and 3 straight x plus straight C
end style

Solution 31

begin mathsize 12px style Here comma space open parentheses straight x squared minus 1 close parentheses dy over dx plus 2 open parentheses straight x plus 2 close parentheses straight y equals 2 left parenthesis straight x plus 1 right parenthesis
dy over dx plus fraction numerator 2 left parenthesis straight x plus 2 right parenthesis over denominator open parentheses straight x squared minus 1 close parentheses end fraction straight y equals fraction numerator 2 left parenthesis straight x plus 1 right parenthesis over denominator open parentheses straight x squared minus 1 close parentheses end fraction
It space is space straight a space linear space differential space equation. space Comparing space it space with comma
dy over dx plus py equals straight Q
straight P equals fraction numerator 2 left parenthesis straight x plus 2 right parenthesis over denominator straight x squared minus 1 end fraction comma straight Q fraction numerator 2 left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x squared minus 1 right parenthesis end fraction
straight I. straight F. space equals straight e to the power of integral pdx end exponent
space space space space space space space space space space equals straight e to the power of 2 integral fraction numerator left parenthesis straight x plus 2 right parenthesis over denominator open parentheses straight x squared minus 1 close parentheses end fraction dx end exponent
space space space space space space space space space space equals straight e to the power of 2 integral fraction numerator 2 straight x over denominator straight x squared minus 1 end fraction dx plus 4 integral fraction numerator 1 over denominator straight x squared minus 1 end fraction dx end exponent
space space space space space space space space space space equals straight e to the power of log open vertical bar straight x squared minus 1 close vertical bar plus 4 straight x 1 half log open vertical bar fraction numerator straight x minus 1 over denominator straight x plus 1 end fraction close vertical bar end exponent
space space space space space space space space space space equals straight e to the power of log open vertical bar straight x squared minus 1 close vertical bar plus log open vertical bar fraction numerator straight x minus 1 over denominator straight x plus 1 end fraction close vertical bar squared end exponent
space space space space space space space space space space equals straight e to the power of log fraction numerator open parentheses straight x squared minus 1 close parentheses left parenthesis straight x minus 1 right parenthesis squared over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction end exponent
straight I. straight F space space space space space space equals fraction numerator open parentheses straight x plus 1 close parentheses left parenthesis straight x minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis squared over denominator left parenthesis straight x plus 1 right parenthesis squared end fraction
space space space space space space space space space space space space equals fraction numerator left parenthesis straight x minus 1 right parenthesis cubed over denominator left parenthesis straight x plus 1 right parenthesis end fraction
Solution space of space the space equation space is space given space by comma
4 cross times left parenthesis straight I. straight F right parenthesis equals integral straight Q cross times left parenthesis straight I. straight F right parenthesis dx plus straight c
fraction numerator straight y left parenthesis straight x minus 1 right parenthesis cubed over denominator left parenthesis straight x plus 1 right parenthesis end fraction equals 2 straight x squared over 2 minus 6 straight x minus 8 space log open vertical bar straight x plus 1 close vertical bar plus straight c
fraction numerator straight y left parenthesis straight x minus 1 right parenthesis cubed over denominator left parenthesis straight x plus 1 right parenthesis end fraction equals straight x squared minus 6 straight x plus 8 log open vertical bar straight x plus 1 close vertical bar plus straight c
straight y equals fraction numerator straight x plus 1 over denominator left parenthesis straight x minus 1 right parenthesis cubed end fraction open square brackets straight x squared minus 6 straight x minus 8 space log open vertical bar straight x plus 1 close vertical bar plus straight c close square brackets end style

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36(i)

begin mathsize 12px style Here comma space dy over dx plus 3 straight y equals straight e to the power of mx
It space is space straight a space linear space differential space equation. space Comparing space it space with comma
dy over dx plus py equals straight Q
straight P equals 3 comma space straight Q equals straight e to the power of mx
straight I. straight F. equals straight e to the power of integral pdx end exponent
equals straight e to the power of integral 3 dx end exponent
equals straight e to the power of 3 straight x end exponent
Solution space of space the space equation space is space given space by comma
straight y cross times open parentheses straight I. straight F close parentheses equals integral space straight Q space cross times space open parentheses straight I. straight F. close parentheses dx space plus space straight c
straight y open parentheses straight e to the power of 3 straight x end exponent close parentheses space equals space integral straight e to the power of mx space straight e to the power of 3 straight x end exponent space dx space plus space straight c
equals space integral space straight e to the power of open parentheses straight m plus 3 close parentheses straight x end exponent space dx space plus space straight c
straight y open parentheses straight e to the power of 3 straight x end exponent close parentheses equals fraction numerator straight e to the power of open parentheses straight m plus 3 close parentheses straight x end exponent space dx over denominator open parentheses straight m plus 3 close parentheses end fraction plus straight c end style

Solution 36(ii)

Solution 36(iii)

Solution 36(iv)

Solution 36(v)

Solution 36(vi)