5 ms·
> I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all. This may be a b
by Perseids 13y ago
> I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.
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 :).
- ced 13y agoevery ODE that has one and only one solution has an exact solution 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?
- Perseids 13y agoI just stated the definition of having one and only one solution, yes :). That was kind of the point I was making. I originally intended to write a statement specific to ODE, but was then to lazy to look the specifics up ^^.
- joe_the_user 13y agoAh, 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.