Question

Find the general solution of y"-2y'-3y=3e^2t.

Answer 1

first off, solve for homogeneous part to get \[y_h= C_1e^{3t}+C_2e^{-t}\]
second, solve for partial solution by looking for \[y_p= Ae^{2t}\]
last, combine them to get \[y = y_h+y_p= C_1e^{3t}+C_2e^{-t}-e^{2t}\]