Differential equations finding coefficients

Question

Find a particular solution to the following equations by trying a solution of the given form and solve for coefficients.

dy/dt + y = 1+2t^2. y(t)=A + Bt + Ct^2

I found the solution to be y= 5 + 2t^2 - 4t + Ae^-t. How do I go about solving for coefficients.

Answer 1

dy/dt = B + 2Ct

So

B + 2Ct + A + Bt + Ct² = 1 + 2t²

Coefficients of t² : C = 2, so C=2

Coefficients of t : 2C + B = 0 (as no term in t on right hand side), so B = -2C = -4

Constants : B + A = 1, so A = 1-B = 5

So y = 5 -4t + 2t²