I've been studying on my own for a while now. I started with arithmetic and now I'm messing around with differential equations. Sometimes the work is pretty dry and I'm just collecting techniques. Sometimes it's fun and I can see a bigger picture and ask a lot of 'what if...?' questions.
When I started differential equations, it was a lot of fun. I started from the perspective of growth equations, and I understood what I was writing actually meant. Then I learned a bit about 2nd-order DEs.
Now I'm at a 'where am I going with this?' stage. I've hit this wall before, but thought I'd post for some guidance. I don't know anyone else who's interested in math in real life.
I think the general route is 1st order DEs, 2nd order, higher order DEs, non-homogenous DEs, Laplace, Fourier.
What I want is to know as much as any decent graduate would know. But I don't just want to be sucking up techniques, though it's important. I want to be able. So what am I asking?
If you had to put together a set of x questions that, when answered, would show mathematical ability up to graduate level, what would those questions be?
Not only would these questions be a test, but they would also be a guide. Answering each question would be an invitation to explore a certain area of mathematics. The questions wouldn't be shibboleth questions, in other words.
Sorry for the rambling. It's late and I'm tired.