Question

Solve the differential equation $\frac{dy}{dx}+2y=\sin x$.

Answer 1

$\frac{dy}{dx}+2y=\sin x......\left(1\right)$
$P=2,Q=\sin x$ $I.F=e^{\int 2dx}$
On multiplying $e^{2x}$ in equation (1)
$e^{2x}\frac{dy}{dx}+2y{e}^{2x}=e^{2x}\sin x$
$\frac{d}{dx}\left(y{e}^{2x}\right)=e^{2x}\sin x$
On integrating
$y{e}^{2x}=\int e^{2x}\sin xdx+c.....\left(2\right)$
$\int e^{2x}\sin xdx=\frac{e^{2x}}{5}[2\sin x-\cos x]$
$\therefore$ equation (2)
$y{e}^{2x}=\frac{e^{2x}}{5}[2\sin x-\cos x]+c$
On multiplying $e^{-2x}$
$y=\frac{1}{5}[2\sin x-\cos x]+c{e}^{-2x}$