Question

Find the general solution of the differential equation:

$\frac{dy}{dx}=\frac{y}{x}+\sin \left(\frac{y}{x}\right)$

Answer 1

We have

$\frac{dy}{dx}=\frac{y}{x}+sin\left(\frac{y}{x}\right)$ _____ (1)

put y=vx

differentiating

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

Now (1) becomes

$v+x\frac{dv}{dx}=\frac{vx}{x}+sin\left(\frac{vx}{X}\right)$

$v+x\frac{dv}{dx}=v+sinv$

$x\frac{dv}{dx}=sinv$

$\frac{dv}{sinv}=\frac{dx}{x}$

integrating both sides, we get

$\int cosecvdv=\int \frac{dx}{x}$

$log|cosecv-cotv|=logx+logc$

put v value

$log|cosec\frac{y}{x}-cot\frac{y}{x}|=log|cx|$

$cosec\left(\frac{y}{x}\right)-cot\left(\frac{y}{x}\right)=cx$

$C=\frac{cosec\left(\frac{y}{x}\right)-cot\left(\frac{y}{x}\right)}{x}$