I had some trouble on an exam recently with the particular solution of a Nonhomogeneous Second Order Ordinary Differential Equation.
So, the problem was: \begin{cases} y''+4y=\sec(2t) \\ y(0)=y'(0)=0 \\ \end{cases}
Solve the Initial Value Problem, and identify the domain of definition of the solution function.
I was able to solve for the homogeneous solution of $Y_H=c_1\sin(2t) + c_2\cos(2t)$ but I had some serious trouble with getting the particular solution.
The hint said to use either Variation of Parameters or the Green Function method. I was not able to do either of these to completion in the time allotted, and I would appreciate someone helping explain a bit about how to apply those to this equation.
EDIT: Someone below has explained how to apply Variation of Parameters to this pretty well. Can someone help explain Greens Function method to me?