Question

For each of the differential equations given in exercises $1$ to $12$.find the general solution.
$\frac{dy}{dx}+2y=\sin x$

Answer 1

$\frac{dy}{dx}+2y=\sin x$

$If=e^{\int 2dx}=e^{2x}$

$\text{General Solution,}$

$y.If=\int \left(If\right)\sin x+c$

$y.e^{2x}=\int e^{2x}\sin x+c$

$y{e}^{2x}=\sin x\frac{e^{2x}}{2}-\int \cos x\frac{e^{2x}}{2}+c$

$I=\sin x\frac{e^{2x}}{2}-\cos x\frac{e^{2x}}{4}-\int \sin x\frac{e^{2x}}{4}dx$

$I=\sin x\frac{e^{2x}}{2}-\cos x\frac{e^{2x}}{4}-\frac{I}{4}$

$\frac{5I}{4}=\frac{e^{2x}}{4}\left[2\sin x-\cos x\right]$

$I=\frac{e^{2x}}{5}\left(2\sin x-\cos x\right)$

$y{e}^{2x}=\frac{e^{2x}}{5}\left(2\sin x-\cos x\right)+c$

$y=\frac{2\sin x-\cos x}{5}+c{e}^{-2x}$