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RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 22 - Differential Equations.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 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 21


Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

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.

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 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)

Solution 16(v)

Solution 16(vi)

Solution 16(vii)

Solution 16(viii)

Solution 16(ix)

Solution 16(x)

Solution 17

Solution 18

Solution 19

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 22.3

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

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

Solution 21(i)

Solution 21(ii)

Solution 21(iii)

  

Solution 21(iv)

Solution 21(v)

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 22.4

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 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

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25


Solution 26

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 22.6

Solution 1

Solution 2

Solution 3

Solution 4

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 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

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

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

Solution 43

Solution 44

Solution 45(i)

Solution 45(ii)

Solution 45(iii)

Solution 45(iv)

Solution 45(v)

Solution 45(vi)

Solution 45(vii)

x y fraction numerator d y over denominator d x end fraction equals open parentheses x plus 2 close parentheses open parentheses y plus 2 close parentheses comma space y open parentheses 1 close parentheses equals minus 1
fraction numerator y d y over denominator open parentheses y plus 2 close parentheses end fraction equals fraction numerator open parentheses x plus 2 close parentheses over denominator x end fraction d x
integral open parentheses 1 minus fraction numerator 2 over denominator y plus 2 end fraction close parentheses d y equals integral open parentheses 1 plus 2 over x close parentheses d x
y minus x minus 2 log left parenthesis y plus 2 right parenthesis minus 2 log x equals c
P u t space x equals 1 comma space y equals minus 1
minus 1 minus 1 minus 2 log left parenthesis minus 1 plus 2 right parenthesis minus 2 log 1 equals c
rightwards double arrow minus 2 equals c
T h u s comma space w e space h a v e
y minus x minus 2 log left parenthesis y plus 2 right parenthesis minus 2 log x equals minus 2

Solution 45(viii)

  

Solution 45(ix)

  

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

Solution 53

Solution 54

Solution 55

Solution 56

Solution 57

..

Solution 58

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 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

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 22.9

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

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

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24


Solution 25

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)

Solution 36(iv)

Solution 36(v)


Solution 36(vi)


Solution 36(vii)

Solution 36(viii)

  

Solution 36(ix)

Solution 37

Solution 38

Solution 39

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 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

Solution 31


Solution 32

Solution 33

Solution 34

Solution 35

Solution 36(i)

Solution 36(ii)

Solution 36(iii)

Solution 36(iv)

Solution 36(v)

Solution 36(vi)

Solution 36(vii)

Solution 36(viii)

Solution 36(ix)

Solution 36(x)

Solution 36(xi)

Solution 36(xii)

Solution 37(i)

Solution 37(ii)

Solution 37(iii)

Solution 37(iv)

Solution 37(v)

Solution 37(vi)

Solution 37(vii)

Solution 37(viii)

Solution 37(ix)

  

Solution 37(x)

  

Solution 37(xi)

  

Solution 37(xii)

  

Solution 38

Solution 39

Solution 40

Solution 41

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 22.11

Solution 1

Solution 2

Solution 3

Solution 4

L e t space C space b e space t h e space c o u n t space o f space b a c t e r i a space a t space a n y space t i m e space t.
I t space i s space g i v e n space t h a t
fraction numerator d C over denominator d t end fraction infinity C
rightwards double arrow fraction numerator d C over denominator d t end fraction equals lambda C comma space w h e r e space lambda space i s space a space c o n s tan t space o f space p r o p o r t i o n a l i t y
rightwards double arrow fraction numerator d C over denominator C end fraction equals lambda d t
rightwards double arrow integral fraction numerator d C over denominator C end fraction equals lambda integral d t
rightwards double arrow log C equals lambda t plus log K.... left parenthesis 1 right parenthesis
I n i t i a l l y comma space a t space t equals 0 comma space C equals 100000
T h u s comma space w e space h a v e comma
log 100000 equals lambda cross times 0 plus log K.... left parenthesis 2 right parenthesis
rightwards double arrow log 100000 equals log K.... left parenthesis 3 right parenthesis
A t space t equals 2 comma space C equals 100000 plus 100000 cross times 10 over 100 equals 110000
T h u s comma space f r o m space left parenthesis 1 right parenthesis comma space w e space h a v e comma
log 110000 equals lambda cross times 2 plus log K.... left parenthesis 4 right parenthesis
S u b t r a c t i n g space e q u a t i o n space left parenthesis 2 right parenthesis space f r o m space left parenthesis 4 right parenthesis comma space w e space h a v e comma
log 110000 minus log 100000 equals 2 lambda
rightwards double arrow log 11 cross times 10000 minus log 10 cross times 10000 equals 2 lambda
rightwards double arrow log fraction numerator 11 cross times 10000 over denominator 10 cross times 10000 end fraction equals 2 lambda
rightwards double arrow log 11 over 10 equals 2 lambda
rightwards double arrow lambda equals 1 half log 11 over 10.... left parenthesis 5 right parenthesis
W e space n e e d space t o space f i n d space t h e space t i m e space apostrophe t apostrophe space i n space w h i c h space t h e space c o u n t space r e a c h e s space 200000.
S u b s t i t u t i n g space t h e space v a l u e s space o f space lambda space a n d space K space f r o m space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 5 right parenthesis 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
log 200000 equals 1 half log 11 over 10 t plus log 100000
rightwards double arrow 1 half log 11 over 10 t equals log 200000 minus log 100000
rightwards double arrow 1 half log 11 over 10 t equals log 200000 over 100000
rightwards double arrow 1 half log 11 over 10 t equals log 2
rightwards double arrow t equals fraction numerator 2 log 2 over denominator log 11 over 10 end fraction space h o u r s

 

Solution 5


Solution 6


Solution 7


L e t space P space b e space t h e space p o p u l a t i o n space o f space t h e space c i t y space a t space a n y space t i m e space t.
I t space i s space g i v e n space t h a t
fraction numerator d P over denominator d t end fraction infinity P
rightwards double arrow fraction numerator d P over denominator d t end fraction equals lambda P comma space w h e r e space lambda space i s space a space c o n s tan t space o f space p r o p o r t i o n a l i t y
rightwards double arrow fraction numerator d P over denominator P end fraction equals lambda d t
rightwards double arrow integral fraction numerator d P over denominator P end fraction equals lambda integral d t
rightwards double arrow log P equals lambda t plus log K.... left parenthesis 1 right parenthesis
I n i t i a l l y comma space a t space t equals 1990 comma space P equals 200000
T h u s comma space w e space h a v e comma
log 200000 equals lambda cross times 1990 plus log K.... left parenthesis 2 right parenthesis
A t space t equals 2000 comma space P equals 250000
T h u s comma space f r o m space left parenthesis 1 right parenthesis comma space w e space h a v e comma
log 250000 equals lambda cross times 2000 plus log K.... left parenthesis 3 right parenthesis
S u b t r a c t i n g space e q u a t i o n space left parenthesis 2 right parenthesis space f r o m space left parenthesis 3 right parenthesis comma space w e space h a v e comma
log 250000 minus log 200000 equals 10 lambda
rightwards double arrow log 4 over 5 equals 10 lambda
rightwards double arrow lambda equals 1 over 10 log 4 over 5.... left parenthesis 4 right parenthesis
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 lambda space f r o m space e q u a t i o n space left parenthesis 4 right parenthesis 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
log 200000 equals 1990 cross times 1 over 10 log 4 over 5 plus log K
rightwards double arrow log K equals log 200000 minus 199 cross times log 4 over 5.... left parenthesis 5 right parenthesis space space
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 lambda comma space log K space a n d space t equals 2010 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
log P equals open curly brackets 1 over 10 log 4 over 5 close curly brackets 2010 plus log 200000 minus 199 cross times log 4 over 5
rightwards double arrow log P equals log open curly brackets 4 over 5 close curly brackets to the power of 201 plus log open parentheses 200000 cross times open parentheses 5 over 4 close parentheses to the power of 199 close parentheses
rightwards double arrow P equals open curly brackets 4 over 5 close curly brackets to the power of 201 cross times 200000 cross times open parentheses 5 over 4 close parentheses to the power of 199
rightwards double arrow P equals open parentheses 5 over 4 close parentheses squared cross times 200000 equals 25 over 16 cross times 200000 equals 312500

 

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

Solution 31

Solution 32

Solution 33

Solution 34

RD Sharma Solution for Class 12 Science Mathematics Chapter 22 - Differential Equations Page/Excercise 22RE

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 2

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 3(v)

Solution 3(vi)

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

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 space y equals a e to the power of 2 x end exponent plus b e to the power of minus 3 x end exponent plus c e to the power of x
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
fraction numerator d y over denominator d x end fraction equals 2 a e to the power of 2 x end exponent minus 3 b e to the power of minus 3 x end exponent plus c e to the power of x.... left parenthesis 1 right parenthesis
rightwards double arrow 7 fraction numerator d y over denominator d x end fraction equals 14 a e to the power of 2 x end exponent minus 21 b e to the power of minus 3 x end exponent plus 7 c e to the power of x.... left parenthesis 2 right parenthesis
D i f f e r e n t i a t i n g space e q u a t i o n space left parenthesis 1 right parenthesis 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
fraction numerator d squared y over denominator d x squared end fraction equals 4 a e to the power of 2 x end exponent plus 9 b e to the power of minus 3 x end exponent plus c e to the power of x.... left parenthesis 3 right parenthesis
A g a i n space 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
fraction numerator d cubed y over denominator d x cubed end fraction equals 8 a e to the power of 2 x end exponent minus 27 b e to the power of minus 3 x end exponent plus c e to the power of x.... left parenthesis 4 right parenthesis
N o w space c o n s i d e r space t h e space f o l l o w i n g space e x p r e s s i o n
fraction numerator d cubed y over denominator d x cubed end fraction minus 7 fraction numerator d y over denominator d x end fraction plus 6 y
equals 8 a e to the power of 2 x end exponent minus 27 b e to the power of minus 3 x end exponent plus c e to the power of x minus 14 a e to the power of 2 x end exponent plus 21 b e to the power of minus 3 x end exponent minus 7 c e to the power of x plus 6 open parentheses a e to the power of 2 x end exponent plus b e to the power of minus 3 x end exponent plus c e to the power of x close parentheses
equals 8 a e to the power of 2 x end exponent minus 27 b e to the power of minus 3 x end exponent plus c e to the power of x minus 14 a e to the power of 2 x end exponent plus 21 b e to the power of minus 3 x end exponent minus 7 c e to the power of x plus 6 a e to the power of 2 x end exponent plus 6 b e to the power of minus 3 x end exponent plus 6 c e to the power of x
equals 0
T h u s comma space t h e space r e q u i r e d space d i f f e r e n t i a l space e q u a t i o n space c o r r e s p o n d i n g space t o space
space y equals a e to the power of 2 x end exponent plus b e to the power of minus 3 x end exponent plus c e to the power of x space i s
fraction numerator d cubed y over denominator d x cubed end fraction minus 7 fraction numerator d y over denominator d x end fraction plus 6 y equals 0

 

Solution 16

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Solution 31

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Solution 50