Second order linear differential equation

Question

I have to teach the following methods to my juniors at college to solve differential equations:

1) partial fractions

2) reduction of order

3) variation of parameter

4) power series

5) green's function

I was thinking of taking a non-trivial second order linear differential equation that can be solved by all the above methods. Please help me with some examples.

Answer 1

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.