Undetermined coefficients guessing particular solution's shape

I try to find general solution for: $$y'' - y = 8 t e^t$$ I find comp. as $(D-1)(D+1)=0$ and general solution of it as $y_g= C_1 e^t + C_2 e^-t$ at this point.

I know particular solution shape is: $$y= A t e^t+ B t^2 e^t$$ But what is the way to find it? In other words how can I guess that?

We are using the Method of Undetermined Coefficients.

The homogeneous solution is:

$$y_h(t) = c_1 e^t + c_2 e^{-t}$$

Since we have an $e^t$ in the homogeneous, the method has us multiply by an extra $t$ term, so we try:

$$y_p(t) = t(a e^t + b t e^t)$$

You can see a good discussion of the reasons and more examples on the linked web site, particularly example $10$. It shows that you try several approaches until you find one that works. After a little experience, you see a better approach for a choice as there is a pattern.

• Thanks alot link is very helpful – Palindrom Nov 24 '13 at 20:26
• Needs a TU! + 1 – Namaste Nov 24 '13 at 23:59