Question

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

Answer 1

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

Let $y=vx$

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

$\Rightarrow x(v+x\frac{dv}{dx})-vx+x\sin v=0$

$\Rightarrow xv+x^2\frac{dv}{dx}-vx+x\sin v=0$

$\Rightarrow x^2\frac{dv}{dx}=-x\sin v$

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

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

Integrating both sides, we get

$\Rightarrow \int \frac{dv}{\sin v}=\int \frac{-dx}{x}$

$\Rightarrow \int \csc vdv=\int \frac{-dx}{x}$

$\Rightarrow \log |\csc v-\cot v|+\log \left|x\right|=\log c$

$\Rightarrow \log x|\csc v-\cot v|=\log c$

$\Rightarrow x(\csc v-\cot v)=c$

$\Rightarrow x(\frac{1}{\sin v}-\frac{\cos v}{\sin v})=c$

$\Rightarrow x\frac{1-\cos v}{\sin v}=c$

$\Rightarrow x(1-\cos v)=c\sin v$

$\Rightarrow x(1-\cos \left(\frac{y}{x}\right))=c\sin \left(\frac{y}{x}\right)$ where $v=\frac{y}{x}$