How Do You Solve Nonhomogeneous Equations with Undetermined Coefficients?

[cue928]

I'm working on undetermined coefficients for nonhomogeneous equations. I have an equation that is equal to 3xe^x. For an earlier one that was equal to e raised to 8x, I used Ae^8x; but on this one, obviously that won't work. I don't know what I should be looking at. I tried Axe^x but no dice on that one either. I know things are supposed to cancel but for two terms multiplied together, how does that work for cancelling purposes? Is it really just a guessing game as to what the form of the particular solution should be?

 

[vela]

You want to use y_p(x) = (Ax+B)e^x. I'm assuming that these terms are independent of your homogeneous solutions.

 

[cue928]

They are, but how did you know how to use that formula? Our book suggests determining it is an educated guess but another book I use as a reference seems to suggest there are standard formulas. What should I be looking at in determining what to use? Thanks!

 

[vela]

You look at the forcing term and "guess" the form from that. Wikipedia summarizes the cases:

http://en.wikipedia.org/wiki/Method...ents#Typical_forms_of_the_particular_solution

In this problem, the forcing term was 3xe^x, which is of the form p(x)e^x where p(x) is a degree-1 polynomial, so your trial solution should be q(x)e^x where q(x)=Ax+B is a polynomial of the same degree.