Solution to nonhomogenous DE using Method of Undetermined Coefficients

[DWill]

Homework Statement

Find the general solution of the given differential equation:

y'' + 9y = (t^2)(e^3t) + 6

Homework Equations

The Attempt at a Solution

I want to first find a particular solution using the method of undetermined coefficients, but I'm not sure what I should "guess" for the form of Y(t). I learned that if g(t) is some exponential function such as e^2t, then I should assume the form looks like Y(t) = Ae^2t, assuming no duplication with solution to the homogenous equation. This doesn't seem to be much harder, but the t^2 and the +6 is confusing me a bit. Any help for how I can get started? Thanks!

 

[HallsofIvy]

In general, if the right hand side is a polynomial, you will require a polynomial of the same degree- but since the coefficients are unknown, even powers that do not appear in the right hand side may be required in the solution. Try $$(At^2+ Bt+ C)e^{3t}$$. For the constant "6" try a constant. You can do that separately using a solution of the form A, and then add them, or do them together with $$(At^2+ Bt+ C)e^{3t}+ D$$