Find a particular solution to the differential equation y′′+2y′+y=2t2−t+3e−2t. y′′+2y′+y=2t2−t+3e−2t.

The characteristic equation is



so the characteristic solution is



For the particular solution, we can try looking for a solution of the form





Substituting into the ODE, we have





So the general solution to the ODE is