Finding a Homogeneous D.E. that has a particular solution

[Alex Ruiz]

Homework Statement

The problem reads:

Find a homogeneous linear differential equation with constant coefficients that has the following particular solution:

yp = e^(-t) + 2te^(t) + t^(2)e^(t) - sin(3t)

Express your equation in differential operator form. (Hint: What annihilators would you need to annihilate everything in this particular solution?)

Homework Equations

The annihilators that I put for each term were:

((D-1)^3) for (2te^(t) + t^(2)e^(t))

(D+1) for (e^(-t))

(D^(2) + 9) for (sin(3t))

The Attempt at a Solution

After plugging in I got:

(D+1)((D-1)^3)(D^(2) + 9)y = 0

I was wondering is this the correct answer since he stated to express the equation in differential operator form? Or is there another step I'm missing. If I'm doing this completley wrong I would greatly appreciate any clarity for this problem. Thank you in advance for any help.

 

[RUber]

This looks right to me. There may be some other simplification you could do, but I don't see anything wrong with what you have.