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

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

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

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