Question

Solve: $\left(x\sin \frac{y}{x}\right)dy=(y\sin \frac{y}{x}-x)dx$.

Answer 1

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

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

let $y=vx$

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

$v+x\frac{dv}{dx}=\frac{yx\sin v-x}{x\sin v}$

$v+x\frac{dv}{dx}=\frac{x(v\sin v-1)}{x\sin v}$

$v+x\frac{dv}{dx}=v-\frac{1}{\sin v}$

$-\sin vdv=\frac{dx}{x}$

Integrationg both sides we get

$\int -\sin vdv=\int \frac{dx}{x}$

$\cos v=lnx+c$

$\cos \frac{y}{s}=lnx+c$