In each of the following examples verify that the given function is a solution of the corresponding  differntial equation. 

Asked by aadeshpdabre | 24th May, 2016, 11:00: AM

Expert Answer:

1. space
C o n s i d e r space t h e space e q u a t i o n comma space y equals c subscript 1 sin x plus c subscript 2 cos x space space space... 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 comma space w e space h a v e comma
fraction numerator d y over denominator d x end fraction equals c subscript 1 cos x minus c subscript 2 sin x space space space... left parenthesis 2 right parenthesis
D i f f e r e n t i a t i n g space o n c e space a g a i 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 squared y over denominator d x squared end fraction equals negative c subscript 1 sin x plus c subscript 2 cos x space space space space.... left parenthesis 3 right parenthesis

A d d i n g space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma
fraction numerator d squared y over denominator d x squared end fraction plus y equals c subscript 1 sin x minus c subscript 2 c o s x minus c subscript 1 sin x plus c subscript 2 cos x
rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction plus y equals 0

2.
G i v e n space t h a t space y space cross times s e c x equals tan x plus c
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 comma space w e space h a v e comma
fraction numerator d y over denominator d x end fraction cross times s e c x plus y s e c x cross times tan x equals s e c squared x
D i v i d i n g space t h e space e q u a t i o n space b y space s e c x comma space w e space h a v e comma
fraction numerator d y over denominator d x end fraction plus y cross times t a n x equals s e c x

3.
G i v e n space t h a t space x y equals log y plus c
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 comma space w e space h a v e comma
fraction numerator d y over denominator d x end fraction cross times x plus y equals 1 over y fraction numerator d y over denominator d x end fraction
rightwards double arrow fraction numerator d y over denominator d x end fraction cross times x minus 1 over y fraction numerator d y over denominator d x end fraction equals negative y
rightwards double arrow 1 over y fraction numerator d y over denominator d x end fraction minus fraction numerator d y over denominator d x end fraction cross times x equals y
rightwards double arrow fraction numerator 1 minus x y over denominator y end fraction cross times fraction numerator d y over denominator d x end fraction equals y
rightwards double arrow open parentheses 1 minus x y close parentheses cross times fraction numerator d y over denominator d x end fraction equals y squared
rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator y squared over denominator open parentheses 1 minus x y close parentheses end fraction

Answered by Vimala | 24th May, 2016, 05:42: PM