- #1
Hercuflea
- 596
- 49
Homework Statement
I found everything except step #5. Please tell me if I am correct
Find a particular solution to
(D - 1)(D[itex]^{2}[/itex] + 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[itex]_{p}[/itex]
5) Plug in y[itex]_{p}[/itex] 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[itex]^{2}[/itex]+1 annihilates cos(t)
2) Set A = [e[itex]^{t}[/itex], e[itex]^{-6t}[/itex], e[itex]^{2t}[/itex]]
3) Set B = [e[itex]^{t}[/itex], e[itex]^{-6t}[/itex], e[itex]^{2t}[/itex], cos(t), sin(t)]
4) B-A = [cos(t), sin(t)]
So y[itex]_{p}[/itex] = c[itex]_{1}[/itex]cos(t) + c[itex]_{2}[/itex]sin(t)
5) Expanded equation:
(D[itex]^{3}[/itex]+3D[itex]^{2}[/itex]-16D+12)(c[itex]_{1}[/itex]cos(t) + c[itex]_{2}[/itex]sin(t)) = cos(t)
After fully expanding using FOIL,
9c[itex]_{1}[/itex] - 17 c[itex]_{2}[/itex] = 1
17c[itex]_{1}[/itex] + 9c[itex]_{2}[/itex] = 0
I used matrix transformations to find
c[itex]_{1}[/itex] = [itex]9/370[/itex] and
c[itex]_{2}[/itex] = [itex]-17/370[/itex]
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.
Last edited: