Question

Solve the differential equation: $y\frac{dy}{dx}=\frac{x}{e^y}$

Answer 1

Given, $y\frac{dy}{dx}=\frac{x}{e^y}$
$\Rightarrow \int y\cdot e^{y}dy=\int xdx$
$\Rightarrow y\cdot e^y-\int 1\cdot e^{y}dy=\frac{x^2}{2}+c$
$\Rightarrow y\cdot e^y-e^y=\frac{x^2}{2}+c$
$\Rightarrow (y-1)e^y=\frac{x^2}{2}+c$.