Question

Solve: $\frac{dy}{dx}=e^x-y$.

Answer 1

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

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

or, $e^x\frac{dy}{dx}+e^{x}y=e^{2x}$ [Multiplying both sides with $e^x$]

or, $\frac{d}{dx}\left(y{e}^x\right)=e^{2x}$

Integrating we have

$y{e}^x=\frac{e^{2x}}{2}+c$ [$c$ being integrating constant]

or, $y=\frac{e^x}{2}+c{e}^{-x}$