Question

Solve

$(x\frac{dy}{dx}-y)e^{\frac{y}{x}}=x^2\cos x$

Answer 1

Given,

$(x\frac{dy}{dx}-y)e^{\frac{y}{x}}=x^2\cos x$

$(x\frac{dy}{dx}-y)=\frac{x^2\cos x}{e^{\frac{y}{x}}}$

$x\frac{dy}{dx}=\frac{x^2\cos x}{e^{\frac{y}{x}}}+y$

$\frac{dy}{dx}=\frac{x\cos x}{e^{\frac{y}{x}}}+\frac{y}{x}$

substitute $\frac{y}{x}=v$

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

$x\frac{dv}{dx}+v=\frac{x\cos x}{e^v}+v$

$x\frac{dv}{dx}=\frac{x\cos x}{e^v}$

$\frac{dv}{dx}=\frac{\cos x}{e^v}$

$e^{v}dv=\cos xdx$

integrating on both sides, we get,

$e^v=\sin x+c$

$e^{\frac{y}{x}}=\sin x+c$