Superposition and variation of parameters

[Aerosion]

Homework Statement

y''+2y'+y = 4t^2 - 3 + (e^-t)/t

of course i evaluated the general soltuion to be c1e^-1t + c2te^-1t

but now how do you do the right part? i tried y=At^2+Bt+c+1/(Dt+E)*e^-t as a solution but after differentiating it twice and putting it into the eqaution i got...

(4e^t/Dt+E)-(4De^t/(Dt+E^2))+(2D^2e^t/(Dt+E)^3)+At^2+4At+Bt+2A+2B+C = 4t^2-3+(e^-t)/t

and i don't know what to do with that. i found a=4 and b=0 and c=-11 but that's about all i did, I'm unsure how to find the rest of the letters to complete the probleme

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Since (1/t) e^-t is NOT one of the solutions one would expect to get as a solution to a homogeneous d.e. with constant coefficients, "undetermined coefficients" will not work. You titled this "superposition and variation of parameters". Have you tried variation of parameters to get a specific solution for the (1/t)e^-t?