Answer to Question #300868 in Differential Equations for Cross

Question

dy/dx+ycotx=5e^cosx

Answer 1

Integrating factor

$\mu(x)=e^{\int\cot xdx}=e^{\ln (\sin x)}=\sin x$

$\int \cot xdx=\int\dfrac{\cos x}{\sin x}dx=\ln(|\sin x|)+C$

$y'\sin x+y\cos x=5e^{\cos x}\sin x$

$d(y\sin x)=5e^{\cos x}\sin xdx$

Integrate

$\int d(y\sin x)=\int 5e^{\cos x}\sin xdx$

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

$y=-5e^{\cos x}\csc x+C\csc x$