Re:What is it like to be a mathematician?
by joe_the_user on Monday October 14, 2013 @12:33AM
(#6545421)
Ah,
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.
every 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?
Re:What is it like to be a mathematician?
by Perseids on Monday October 14, 2013 @04:01AM
(#6545977)
I 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 ^^.
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 :).