Find the general solution of x'1 = 3x1 - x2 + et, x'2 = x1.

Note that if , then , and so we can collapse the system of ODEs into a linear ODE:




which is a pretty standard linear ODE with constant coefficients. We have characteristic equation



so that the characteristic solution is



Now let's suppose the particular solution is . Then



and so



Thus the general solution for is



and you can find the solution by simply differentiating .