Non-homogenous differential Equation

sndoyle1

1. The problem statement, all variables and given/known data
solve:
y""+6y'+9y=e^-3x/x³


2. Relevant equations

y=y_c+y_p


3. 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 [itex]e^{ax}/x^k[/itex] try using [itex]Ae^{-3x}/x[/itex]

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 [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

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.