(20 pts) Indicate the form in which you'd be looking for a solution to
Do not solve the equation.
First write the characteristic equation:
So we get one root: r1 = 0. The equation r3 + 2r2 + 2r + 1 = 0 has an obvious root r2 = -1. It must be divisible by (r + 1). The long division gives
The latter (quadratic) polynomial has two roots: r3, 4 = (-1 ± sqrt(1 - 4))/2 = (-1 ± i·sqrt(3))/2.
Now we have to consider the right hand side one piece at a time:
3et is of the Aet kind with an eye on r = 1 as a root of the characteristic equation. There is none. Therefore, we should try a partial solution in the same form:
2te-t is of the e-t(B + Ct) kind with an eye on r = -1 as a root of the characteristic equation. There's one such root. We therefore should upgrade our guess by a factor of t:
e-tsin(t) is of e-t(Dsin(t) + Ecos(t)) kind with an eye on r = -1 ± i as the roots of the characteristic equation. There are no such roots. We therefore should be looking for a particular solution in the same form: