Question

Find the particular solution of the differential equation dydx+ycos(x)=5cos(x) satisfying the initial condition y(0)=7.

Answer 1

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

With $y(0)=7$, we have

$7=5+Ce^{-\sin 0}\implies 7=5+C\implies C=2$

so that the particular solution is

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