Annihilator method question

[lucas20]

Let A be a constant-coefficient operator with characteristic polynomial Pa(x).
(a) Use the annihilator method to prove that the differential equation A(y) = e^(αx) has a
particular solution of the form:

y = (e^(αx))/Pa(α), if α is not a zero of polynomial Pa(x).


(b) If α is a simple zero of Pa(x) (multiplicity l), prove that the equation A(y) = e^(αx) has the
particular solution:

y = (xe^(αx))/Pa'(x)

ps: if you didn't understand the question text see it on the attachment file.

 

[Ishuda]

Where have you gotten stuck? Assuming you have no way on how to approach the problem, I'll outline how I would start to solve the problem. First, let r_j be the roots of the characteristic equation and a_j be the constant coefficients. Then we have

A(y) = a_n Dⁿy + a_n-1 D^n-1 y... + a₀ y = a_n (D - r_n) (D - r_n-1) ... (D - r₀) y
where Dⁿ is the differential operator (dⁿ / dxⁿ) and a_n is assumed non-zero. Now the annihilator of e^αx is (D - α). So multiply the equations by the annihilator and consider what happens when you have multiple roots.