Question

Solve differential equation $x\frac{dy}{dx}-y+x\sin \left(\frac{y}{x}\right)=0$.

Answer 1

$x\frac{dy}{dx}-y+x\sin \left(\frac{y}{x}\right)=0$
Divide each term by x
$\therefore \frac{dy}{dx}-\frac{y}{x}+\sin \left(\frac{y}{x}\right)=0$
Now put $\frac{y}{x}=v$
$\therefore y=x\cdot v$
$\therefore \frac{dy}{dx}=x\frac{dv}{dx}+v$
$\therefore$ Given homogeneous equation reduces as follows.
$\therefore x\frac{dv}{dx}+v-v+\sin v=0$
$\therefore x=\frac{dv}{dx}-\sin v$
$\therefore \frac{dv}{\sin v}=-\frac{dx}{x}$
$\therefore \int cosecvdv=\int \frac{1}{x}dx$
$\therefore log\left|\tan \left(\frac{v}{2}\right)\right|+logx=logc$
$\therefore log\left|\tan \left(\frac{v}{2}\right)\right|\cdot =logc$
$\therefore \tan \left(\frac{v}{2}\right)\cdot x=c$
$\therefore \tan \left(\frac{y}{2x}\right)\cdot x=c$
Which is necessary solution of given differential equation.