I am in a first year differential equations course, and in class on Friday, the teacher did a problem from the book that I wasn't quite sure how to solve (yet I'm sure has a possibility of showing up on a test!).
The question I have written in my notes is: "create a differential equation that has $ y = C_1e^{-2x} + C_2e^{3x} + C_3xe^{3x} + e^x + x^2 + x $ as its general solution".
How would I go about doing this?
I see that the general solution has $ C_1e^{-2x} + C_2e^{3x} + C_3xe^{3x} $, meaning the characteristic equation for the homogeneous equation ($Y_h$) should have roots $-2, 3, 3$. I guess for that I should make a polynomial which yields these roots?
I also notice that the particular solution ($Y_p$) should be in the form $Ae^x + Bx^2 + Cx$. If we assume the DE I make is in the form $y'' + p(x)y' + q(x)y = f(x)$ - sorry if this isn't standard convention!), then $f(x)$ should contain something like $e^x + x^2 + x$ ?
Perhaps I'm way off.