The solution of the differential equation
` x (dy)/(dx) - y + x sin (y/x) = 0 ` is

Options

A. ` x(1+cos.y/x)= C sin (y/x)`
B. ` (1-cos. y/x)=C sin (x/y)`
C. ` x(1-cos.y/x)=C sin (y/x)`
D. ` (1-cos.y/x)= Cx sin (y/x)`

Answer

c

Explanation

Given ` (dy)/(dx) = y/x -siny/x`
The given differetial equation is homogenous
So put ` y = vx rArr (dy)/(dx) = v + x (dv)/(dx)`
Then, Eq. (i) becomes ` v + x (dv)/(dx) = v - sin v`
` rArr " cosec " v dv = -1/x dx`
On integrating both sides , we get
` int " cosec" v dv = - int (dx)/x`
` rArr log | " cosec" v - cot v | = - log |x | +A `
` :' int " cosec " x dx = log[ "cosec " x - cot x ]`
` rArr log | ("cosec " v - cot v)x| = A`
` rArr | x ( "cosec" v - cot v ) |e^(A)`
` rArr x(1/(sinv)- (cosv)/(sinv))= e^(A)`
` rArr x((1-cosv)/sinv) = e^(A) rArr x (1-cos v) = e^(A) sinV`
` rArr x [ 1-cos (y/x)] = C sin (y/x) [" put "e^(A) = C and y/x = v]`