Non-homogenous differential Equation

[sndoyle1]

Homework Statement

solve:
y""+6y'+9y=e^-3x/x³

Homework Equations

y=y_c+y_p

The Attempt at a Solution

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

- annihilators
- undetermined coefficients
- variation of paramaters.

 

[sandy.bridge]

Since it is in the form $e^{ax}/x^k$ try using $Ae^{-3x}/x$

 

[sndoyle1]

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]

I usually always try the simplest first. This doesn't pertain to this question, but if $Ae^{ax}$ didn't work I would try $Axe^{ax}$, and if that didn't work I would try $Ax^2e^{ax}$. It can be rather tedious for some questions but eventually you start to notice patterns.

 

[HallsofIvy]

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]

I think he accidentally hit the quotation mark key.