These notes are set so that you get to prove the main results by solving smaller problems that when put together give the big result. The answers to the problems are in the videos. You will get the most out of these notes if you do (or try) the problems before looking at the videos.

Higher order constant coefficient equations.



Before going on to higher order equations it is interest to see that the method of undetermined coefficients is useful in finding some integrals that involve doing repeated integration by parts.

Problem: Use the method of undetermined coefficients to find compute \( \displaystyle \int x^3 e^{2x}\,dx\).


Solution.



Problem: Use the method of undetermined coefficients to compute \( \displaystyle \int e^{2x} \cos(3x)\,dx\).


Solution.



Problem: The homogeneous linear differential equation $$ y^{(4)} - 10 y''' + 41 y'' - 76 y + 52 y=0. $$ has as characteristic polynomial $$ P(r) = r^4 - 10 r^3 + 41r^2 - 76 r + 52 = (r-2)^2 (r^2 - 6r + 13). $$

(a) Find the general solution to this equation.

(b) Find the general solution to the inhomogeneous equation $$ y^{(4)} - 10 y''' + 41 y'' - 76 y + 52 y= 32 e^x. $$
Solution.