Question

Find the particular solution of the differential equation \frac{dy}{dx} + y\cos(x) = 5\cos(x)

Answer 1

$\dfrac{\mathrm dy}{\mathrm dx}+y\cos x=5\cos x$
$e^{\sin x}\dfrac{\mathrm dy}{\mathrm dx}+y\cos xe^{\sin x}=5\cos xe^{\sin x}$
$\dfrac{\mathrm d}{\mathrm dx}\left[e^{\sin x}y\right]=5\cos xe^{\sin x}$
$e^{\sin x}y=5\displaystyle\int\cos xe^{\sin x}\,\mathrm dx$
$e^{\sin x}y=5e^{\sin x}+C$
$y=5+Ce^{-\sin x}$