Consider the differential equation ay by cy ekx, where a, b, c, and k are constants. The

Chapter 3, Problem 43

(choose chapter or problem)

Consider the differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=e^{k x}\), where a, b, c, and k are constants. The auxiliary equation of the associated homogeneous equation is

\(a m^{2}+b m+c=0\) .

(a) If k is not a root of the auxiliary equation, show that we can find a particular solution of the form \(y_{p}=A e^{k x}\), where \(A=1 /\left(a k^{2}+b k+c\right)\)

(b) If k is a root of the auxiliary equation of multiplicity one, show that we can find a particular solution of the form \(y_{p}=A x e^{k x}\), where A=1/(2ak+b). Explain how we know that \(k \neq-b /(2 a)\).

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