Question

let L(y)=y'' +a_1 y' + a_2 y where a1,a2 are constants and let p be the characteristic polynomial p(r)=r^2 + a_1r + a2. Compute a particular solution of L(y) = Ae^(alpha x) in case p(alpha)=0. i have the answers but i dont know how they got to it

Answer 1

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Answer 2

if multiplicity of alpha is 1 then phi = (Axe^(alpha x))/p'(x)
if multiplicity of alpha is 2 then phi = (A/2)x^2 e^(alpha x)