Q:

Method of variation of parameter

By method of variation of parameters find the PI of the differential equation 
$$\frac{dy}{dx}+2y=4e^{2x}$$
I have learned how to do the find out the PI of second order differential equation. But, here is first order. I can't find any solution of this. Please help me.
$$\text{I tried to use the Wronskian method like:}$$
$$Let, y=e^{mx}\\
\implies Dy=me^{mx}$$
$$\text{Then, the auxiliary equation is:}\\
m+2=0\\
\implies m=-2$$
$$Then, C.F.=c_1e^{mx}$$
$$P.I. y_p= u(x)e^{mx}$$
Then, there we had to find wronskian. But, I can't get it how to find that...

A:

You don't need the Wronskian here it's a first order DE
$$\frac{dy}{dx}+2y=4e^{2x}$$
Solve the homogeneous equation
$$\frac{dy}{dx}+2y=0$$
$$\ln y =-2\int dx=-2x+c$$
$$y=ce^{-2x}$$
Variation of parameter needs
$$y=c(x)e^{-2x}$$
$$y'=c'(x)e^{-2x}-2e^{-2x}c(x)$$
Plug this in the differential equation and solve for $c(x)$.
$$y'+2y=4e^{2x}$$
$$c'(x)e^{-2x}-2e^{-2x}c(x)+2c(x)e^{-2x}=4e^{2x}$$
Can you take it from here ?

