Consider the differential equation ay'' + by' + cy = e^kx, where a, b, c, and k are constants. The auxiliary equation of the associated homogeneous equation is

am² + bm + c = 0.

(a) If k is not a root of the auxiliary equation, show that we can find a particular solution of the form

y_p = Ae^kx, where A = 1/(ak² + bk + c).

(b) If k is a root of the auxiliary equation of multiplicity one, show that we can find a particular solution of the form y_p = Axe^kx, where A = 1/(2ak + b). Explain how we know that k ≠ b/(2a).

(c) If k is a root of the auxiliary equation of multiplicity two, show that we can find a particular solution of the form y = Ax²e^kx, where A = 1/(2a).