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Non-homogenous differential Equation

by sndoyle1
Tags: differential, equation, nonhomogenous
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sndoyle1
#1
Feb27-13, 07:27 PM
P: 6
1. The problem statement, all variables and given/known data
solve:
y""+6y'+9y=e-3x/x3


2. Relevant equations

y=yc+yp


3. The attempt at a solution

I found yc=C1e-3x+C2xe-3x
and am having difficulties finding yp. I am wondering which method would be the best to determine yp:

- annihilators
- undetermined coefficients
- variation of paramaters.
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sandy.bridge
#2
Feb27-13, 07:55 PM
P: 778
Since it is in the form [itex]e^{ax}/x^k[/itex] try using [itex]Ae^{-3x}/x[/itex]
sndoyle1
#3
Feb27-13, 09:09 PM
P: 6
Thanks, it worked out. I have a hard time knowing what 'guess' to use for the derivative. How did you know to put it over x instead of x-3? I have a test tomorrow, so I want to make sure that I can do things properly.

sandy.bridge
#4
Feb27-13, 09:37 PM
P: 778
Non-homogenous differential Equation

I usually always try the simplest first. This doesn't pertain to this question, but if [itex]Ae^{ax}[/itex] didn't work I would try [itex]Axe^{ax}[/itex], and if that didn't work I would try [itex]Ax^2e^{ax}[/itex]. It can be rather tedious for some questions but eventually you start to notice patterns.
HallsofIvy
#5
Feb28-13, 08:13 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,558
Is that really a fourth degree equation or is the second '' a typo?

"Undetermined coefficents" works when the right side of the equation is one of the types of solutions you can get as solutions to homogenous differential equations with constant coefficients: exponentials, sine or cosine, and polynomials, as well as combinations of those. That is not the case here. I recommend "variation of parameters".
sandy.bridge
#6
Feb28-13, 12:12 PM
P: 778
I think he accidentally hit the quotation mark key.


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