Partial fractions is not a method for solving differential equations. It is a method for expressing a rational function as a sum of simpler rational functions. As such, it comes up often in the process of inverting the Laplace transform.
Power series work best for initial value problems for linear ODE with polynomial coefficients. You would not want to solve a boundary value problem with a power series. On the other hand, Green's function is best for boundary value problems.
I strongly recommend you to abandon the idea of presenting the aforementioned methods on one example. The main reason we have multiple methods is that they help us overcome different issues that arise in this subject. There are good online resources for ODE that you could use, from the free textbook by Lebl to MIT courseware.