(2xcosy+y2cosx)dx + (2ysinx – x2siny)dy =0 

What is the solution of this differential equation?

Asked by Malavika Umesh | 16th Oct, 2014, 01:48: PM

Expert Answer:

Q u e s t i o n space s h o u l d space b e : I f space t h e space d i f f e r e n t i a l space e q u a t i o n space open parentheses 2 x cos y plus y squared cos x close parentheses d x plus open parentheses 2 y sin x minus x squared sin y close parentheses d y equals space i s space e x a c t comma space s o l v e space i t. S o l u t i o n : 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 open parentheses 2 x cos y plus y squared cos x close parentheses d x plus open parentheses 2 y sin x minus x squared sin y close parentheses d y equals 0 M open parentheses x comma y close parentheses equals 2 x cos y plus y squared cos x ; space N open parentheses x comma y close parentheses equals 2 y sin x minus x squared sin y D i f f e r e n t i a t i n g space t h e space f u n c t i o n s space M space a n d space N space p a r t i a l l y space w i t h space r e s p e c t space t o space y space a n d space x space r e s p e c t i v e l y comma space k e e p i n g space t h e space o t h e r space v a r i a b l e space a s space c o n s tan t comma space w e space h a v e comma M subscript y equals fraction numerator partial differential M over denominator partial differential y end fraction equals minus 2 x sin y plus 2 y cos x ; space N subscript x equals fraction numerator partial differential N over denominator partial differential x end fraction equals 2 y cos x minus 2 x sin y S i n c e space t h e space g i v e n space e q u a t i o n space i s space e x a c t comma space w e space h a v e comma M subscript y equals N subscript x. T o space s o l v e space t h e space e x a c t space e q u a t i o n space w e space s h o u l d space f i n d space f open parentheses x comma y close parentheses : fraction numerator partial differential f over denominator partial differential x end fraction equals M open parentheses x comma y close parentheses equals minus 2 x sin y plus 2 y cos x rightwards double arrow f open parentheses x comma y close parentheses equals minus x squared sin y plus 2 y sin x plus g open parentheses y close parentheses.... left parenthesis 1 right parenthesis N o w space d i f f r e n t i a t i n g space t h e space a b o v e space f u n c t i o n comma space w i t h space r e s p e c t space t o space y comma space w e space h a v e comma fraction numerator partial differential f over denominator partial differential y end fraction equals N open parentheses x comma y close parentheses equals 2 y cos x minus 2 x sin y plus g apostrophe open parentheses y close parentheses... left parenthesis 2 right parenthesis S i m i l a r l y comma space sin c e comma space space N open parentheses x comma y close parentheses equals 2 y sin x minus x squared sin y fraction numerator partial differential f over denominator partial differential y end fraction equals N open parentheses x comma y close parentheses equals 2 y cos x minus 2 x sin y.... left parenthesis 3 right parenthesis N o w space c o m p a r i n g space e q u a t i o n s space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma 2 y cos x minus 2 x sin y plus g apostrophe open parentheses y close parentheses equals 2 y cos x minus 2 x sin y rightwards double arrow g apostrophe open parentheses y close parentheses equals 0 rightwards double arrow g open parentheses y close parentheses equals c o n s tan t equals c T h u s comma space t h e space s o l u t i o n space o f 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 f open parentheses x comma y close parentheses equals minus x squared sin y plus 2 y sin x plus g open parentheses y close parentheses rightwards double arrow f open parentheses x comma y close parentheses equals minus x squared sin y plus 2 y sin x plus c P a r t i a l space d i f f e r e n t i a t i o n space i s space n o t space t h e r e space i n space t h e space c u r r i c u l u m. B u t space w e space h a v e space p r o v i d e d space t h e space s o l u t i o n space t o space e n c o u r a g e space y o u r space c u r i o s i t y.

Answered by Vimala Ramamurthy | 17th Oct, 2014, 10:56: AM