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CBSE Class 12-science Answered

Solve the defferential (x square+ y square) dx+ 2xy dy=0
Asked by Www.talimkhan.com | 30 Jan, 2019, 08:38: PM
answered-by-expert Expert Answer
Given : begin mathsize 12px style open parentheses x squared plus y squared close parentheses d x space plus space 2 x y space d y space equals space 0 end style
begin mathsize 12px style H e n c e space fraction numerator d y over denominator d x end fraction equals negative fraction numerator x squared plus y squared over denominator 2 x y end fraction space.............. left parenthesis 1 right parenthesis sin c e space f u n c t i o n s space i n space n u m e r a t o r space a n d space d e n o m i n a t o r space a r e space h o m o g e n e o u s l e t space u s space s b s t i t u t e space y space equals space v x space space comma space space space h e n c e space fraction numerator d y over denominator d x end fraction space equals space v space plus space x fraction numerator d v over denominator d x end fraction a f t e r space m a k i n g space s u b s t i t u t i o n space f o r space y space a n d space fraction numerator d y over denominator d x end fraction space i n space e q n. left parenthesis 1 right parenthesis comma space a n d space a f t e r space s i m p l i f i c a t i o n x fraction numerator d v over denominator d x end fraction space equals space minus fraction numerator 3 v squared plus 1 over denominator 2 v end fraction space.............. space left parenthesis 2 right parenthesis e q n. left parenthesis 2 right parenthesis space i s space r e w r i t t e n space a s space 1 third fraction numerator 6 v d v over denominator 3 v squared plus 1 end fraction space equals space minus fraction numerator d x over denominator x end fraction space............ left parenthesis 3 right parenthesis i n t e g r a t i n g space b o t h space s i d e s comma space 1 third log left parenthesis 3 v squared plus 1 right parenthesis space equals space minus space log left parenthesis x right parenthesis space plus space C subscript 1 space........ left parenthesis 4 right parenthesis w h e r e space C subscript 1 space i s space c o n s tan t space o f space i n t e g r a t i o n. space L e t space C subscript 1 space equals log space C subscript 2 t h e n space e q n. left parenthesis 4 right parenthesis space i s space w r i t t e n space a s comma space log left parenthesis 3 v squared plus 1 right parenthesis to the power of bevelled 1 third end exponent plus log left parenthesis x right parenthesis space equals log space C subscript 2 h e n c e comma space space log open square brackets x space left parenthesis 3 v squared plus 1 right parenthesis to the power of bevelled 1 third end exponent space close square brackets space space equals space log space C subscript 2 h e n c e comma space x space left parenthesis 3 v squared plus 1 right parenthesis to the power of bevelled 1 third end exponent space equals space C subscript 3 s u b s t i t u t e v space equals space y divided by x space space i n space a b o v e space e q n. comma space space a n d space r a i s e space t o space t h e space p o w e r space 3 comma space a b o v e space e q n. space b e c o m e s x cubed open square brackets 3 y squared over x squared space plus space 1 close square brackets space equals space C o n s tan t space space space space space space o r space space space space space space space space x left parenthesis 3 y squared plus x squared right parenthesis space equals space C o n s tan t end style
Answered by Thiyagarajan K | 31 Jan, 2019, 12:59: PM

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