Method of Undetermined Coefficients

[aliaja]

Hello and thank you for your desire to help. I'm struggling with the method of undetermined coefficients primarily because I struggle with guessing a solution. For example, I've attached a picture of a problem I'm stuck on. I'm not sure what my "guessed solution" would be for g(x)=6xe^(x). I tried Yp=Ae^x and Yp=Axe^x but I cant seem to get a correct answer. So far I know only that after solving the characteristic equation for the initial given problem, I have Y=Ae^(2x)+Be^((-2/5)x). Thank you for any help.

 

[mathmari]

Since \(\displaystyle g(x)=6xe^{x}\), that means that \(\displaystyle g(x) \) is the product of a polynomial and an exponential, so the "guessed solution" is the product of a polynomial of the same degree with the g(x) and an exponential, \(\displaystyle y_{p}=(Ax+B)e^{x} \).