I'm trying to "Justify the Method of Undetermined Coefficients" by using the "Annihilator Method."

To start, I am given a linear, constant coefficient, nonhomogeneous differential equation:

L[y] = g(x)

Where

L[y] = s_{k}*y^{k} + s_{k-1}*y^{k-1} + ... + s_{0}y

[Note: y^{k} means the k'th derivative of y.]

And

g(x) = p_{n}(x)*e^(ax)cos(bx) + q_{m}(x)*e^(ax)cos(bx)

[Note: n and m are the degrees of the polynomials p and q, respectively.]

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Among the things I need to do is:

1. Show that:

A := [(D-a)^2 + b^2]^(N+1)

is an annihilator for g(x), where N := max(n,m)

2. Show that the auxiliary equation associated with AL[y] = 0 is of the form:

s_{k}*[(r - a)^2 + b^2]^(u+N+1)*(r - r_{2u+1})*...*(r - r_{k}) = 0

where u >= 0 is the multiplicity of a +/- ib as roots of the auxiliary equation associated with

L[y] = 0, and r_{2u+1},...,r_{n} are the remaining roots of this equation.

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Here's the help I need:

1. I don't know how to prove that A := [(D-a)^2 + b^2]^(N+1) is an annihilator for g(x). To

"prove" this would take tons of work and dozens of pages worth of writing (I think). I'm not sure

if I'm expected to "show" this is the annihilator by showing all the work or if I'm just expected to

explain why it's true.

2. I know the assumed form of the solutions to AL[y] = 0 is y = e^(rt), so to find the auxiliary

equation I would need to find AL[e^(rt)], but I'm not sure how to plug this in.

3. Also, if you notice, the auxiliary equation has [(r - a)^2 + b^2]^(u+N+1), where

u is the multiplicity of the complex roots. I have no clue how to get that.

Any help in trying to get through this will be greatly appreciated.