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.