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Find a particular solution of the differential equation $-3y'‘-2y’+y=3xe^x$

I'm not sure how to deal with the $x$ and $e^x$ when multiplied together.

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marked as duplicate by DonAntonio, Hurkyl, rschwieb, Henry T. Horton, Qiaochu Yuan Oct 20 '12 at 2:10

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Uh? What's the question here? – DonAntonio Oct 19 '12 at 23:41
What does the method of undetermined coefficients guess for $3xe^x$? – user1038665 Oct 19 '12 at 23:41
This 'question' is meaningless as it stands. Undetermined coefficients of what? – copper.hat Oct 19 '12 at 23:43
@user1038665 We don't know what you're looking at. We don't know what you mean by "guess". You really need to explain things like we were completely strangers. Oh, we are! =) – Pedro Tamaroff Oct 19 '12 at 23:45
Because the method of "lucky guess" is for differential equations and, in fact, for non-homogeneous ordinary ones?! As it stands, the question is meaningless. – DonAntonio Oct 19 '12 at 23:51
up vote 1 down vote accepted

I'll assume you're talking about a differential equation $P(D) y = 3 x e^x$ where $P$ is a polynomial of degree $n$. If $P(1) \ne 0$, there will be a particular solution of the form $(c_1 x + c_2) e^x$ for some constants $c_1$ and $c_2$. If $1$ is a zero of $P$ of multiplicity $k$, then it will be $(c_1 x + c_2) x^k e^x$.

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