Annihilator method of undetermined coefficients.

[Hercuflea]

Homework Statement

I found everything except step #5. Please tell me if I am correct

Find a particular solution to

(D - 1)(D$^{2}$ + 4D - 12)y = cos(t)

using the annihilator approach of the method of undetermined coefficients.

Homework Equations

1) Find annihilator
2) Find A = fundamental set of corresponding homogeneous equation
3) Find B= fundamental set of the annihilated equation.
4) B-A = y$_{p}$
5) Plug in y$_{p}$ to find the coefficients.

The Attempt at a Solution

I am going to skip typing my work for Steps 1-4, because it would take an insane amount of time.

1) D$^{2}$+1 annihilates cos(t)

2) Set A = [e$^{t}$, e$^{-6t}$, e$^{2t}$]
3) Set B = [e$^{t}$, e$^{-6t}$, e$^{2t}$, cos(t), sin(t)]
4) B-A = [cos(t), sin(t)]

So y$_{p}$ = c$_{1}$cos(t) + c$_{2}$sin(t)

5) Expanded equation:
(D$^{3}$+3D$^{2}$-16D+12)(c$_{1}$cos(t) + c$_{2}$sin(t)) = cos(t)

After fully expanding using FOIL,

9c$_{1}$ - 17 c$_{2}$ = 1
17c$_{1}$ + 9c$_{2}$ = 0

I used matrix transformations to find
c$_{1}$ = $9/370$ and
c$_{2}$ = $-17/370$

Am I correct? These solutions seem way too messy compared to what he has given us in the past. In class, he solved Step 5 without actually FOILing the equation, which I did not quite follow, but if I could figure it out it would be much easier than spending several minutes doing monotonous algebra. I know for a fact he will give us an equation like this (with cos(t) and sin(t)) on the final exam because he did the same for the normal test.

 

[LCKurtz]

The way to check if it is correct is to plug the answer back into the equation and see if it works. To save you some time, I let Maple do the grunt work and your answer for $y_p$ is correct. Good work.