One thing I've hated about differential equations is how I need to guess the form of the solution.
e.g. it's easy to show that solutions to a linear constant-coefficient differential equation such as
have the form of (some linear combination of) exponentials:
You can just plug in and show that the equation is satisfiable.
But I feel there is something wrong if I must find the solutions through guess-and-check.
Yet what I was never told, and what I seem to never be able to find from looking online, is how to derive this fact rigorously, when I don't already have the intuition necessary to guess the form of the solution?
Sitting down and working through some math, I've come up with some nonsense that works quite beautifully:
1. Place the homogeneous equation into the following matrix form:
where, for example, we have
2. Drop the arrows and pretend everything is a scalar:
3. Separate the, uh, variables:
How do I derive it properly?