She of course responded that we couldn't be more wrong.
Annoyed the hell out of me until I came across Godel and began to understand maths from a different perspective.
Wish I'd had chance to study under that professor.
This may be a bit theoretical for most readers, but every ODE that has one and only one solution has an exact solution. That the best definition for that function is "the solution to that ODE" does not mean that the function is any less exact than something you can construct out of exp, sin, cos, etc. In fact it would be completely reasonable to define exp as a solution to a differential equation and then to prove afterwards, that exp is equivalent to the usual power series and accidentally also has some other nice properties :).
I spoke too loosely. I should have written "closed-form symbolic solution" rather than "exact solution" (though "exact" is used colloquially to mean this, it's not clear). Yes, all solutions to an ode would be "exact solutions" in the literal sense.
It is worth noting also that an ode having a solution around a point doesn't imply that it can be extended indefinitely - you can hit "singularities" that make extension impossible.
Upvoted, but - is that a particular theorem, or are you just basically stating the definition of "having a solution"? Wouldn't your statement be true of any system of differential equations, not just ODEs?
I originally intended to write a statement specific to ODE, but was then to lazy to look the specifics up ^^.
This is not entirely true. There are many proofs that begin by assuming CH. There are also two ways of interpreting "need CH to prove". One way is that the thing you are trying to prove is equivalent to CH, which is an interesting/useful result. Another way to interpret it is that you can not come up with another way of proving it. In the latter case, the CH based proof justifies that your proposition is not inconsistent, and may even lead you (or others) to proof that it is true regardless of CH. If I recall correctly some statements have been proved by proving the statement when CH is true, and also when CH is false.
And when you're math as a human endeavor, you can also add "provable but not yet proven" and "proven independent"
So, learn some stuff, see how far from black and white higher math can be.
By the time you get into higher mathematics, you get to a place where nearly everything is unknown. I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.
In ways, math and programming have this in common - as the size of the systems being studied increases, a randomly chosen example becomes harder and harder to deal and thus the "art" consists in choosing important and tractable examples out of a world of huge but intractable systems.