Question

$\frac{dy}{dx}+y\cos x=\frac{1}{2}\cos x$.If $=e^{\int \cos x}=e^{\sin x}$

Answer 1

$\frac{dy}{dx}+ycosx=\frac{1}{2}cosx$

It is first order linear differential equation

$P=cosx,Q=\frac{1}{2}\cos x$

$I.F=e^{\int P.dx}=e^{\sin x}$

Solution

$y\times I.F.=\int Q\times I.F.dx+C$

$y{e}^{\sin x}=\int \frac{1}{2}\cos x{e}^{\sin x}dx+C$

$y{e}^{sinx}=\frac{1}{2}\int e^{sinx}cosxdx+C$

Putting $sinx=t$

$cosxdx=dt$

$y{e}^{sinx}=\frac{1}{2}\int e^{t}dt+C$

$y{e}^{sinx}=\frac{1}{2}{e}^t+C$

$2y{e}^{sinx}=e^{sinx}+C$