Question

Solve the differential equation : $\frac{dy}{dx}+y=\cos x-\sin x$

Answer 1

$\frac{dy}{dx}+Py=Q$

$P=1,Q=\cos x-\sin x$

$IF=e^{\int Pdx}=e^{\int 1dx}=e^x$

$y\left(IF\right)=\int Q.IF$

$y{e}^x=\int (\cos x-\sin x)e^{x}dx$

$=\int \cos x{e}^{x}dx-\int \sin x{e}^{x}dx$

$=e^x\cos x-\int -\sin x{e}^{x}dx-\int \sin x{e}^{x}dx$

$=e^x\cos x+C$

$y{e}^x=e^x\cos x+C$