Create a homogeneous DE given a solution I can't figure this one out.
You are given $y'' + xy = e^x$ and told that $y_p$ solves this DE. How can you find a homogeneous DE so that $y_p$ is a solution to your homogeneous DE?

Consider the DE:  y'' + xy = 0 => let  L = (D^2 + x) y where D= d/dx
=> can be written as Ly = 0
Now Consider the original DE: Ly = e^x => (D - 1)Ly = 0;
Thus you have your Homogeneous DE: (D - 1)Ly = 0
