Enquiry

[DrKareem]

Is it true that if we pice a DE from the set of all DE, the probability to solve it is zero?

[DrKareem]

hmm, made so many typoes :P

here's the real question:

Is it true that if you pick a random Differential Equation from the set of all Differential Equations, the probability to solve it is zero?

[matt grime]

The answer is: pretty much yes. The differential equations that we can solve explicitly and algebraically consitutite practically none of the plethora of those that are out there. Strictly speaking we cannot even solve the average linear homogenous equation in n unknowns as that is as hard as factorizing a degree n polynomial. However we know solutions exist, even though we can't find them.

Of course you need to define your probability measure on the space of things properly for there to be even the hint of a proper answer, and for you to say if you accept numerical methods as a solution, but loosely speaking no we can't solve them. There is peano's criterion for deciding the existence of a solution (but I don't remember when it applies, ie to which class of equations).

[HallsofIvy]

Technically it depends upon what probability function you are using to "pick" the de (matt grime's "probability measure") but under just about any reasonable measure, yes, the set of "solvable des" has measure 0.

I'm not sure about "Peano" but Poisson's "existance and uniqueness theorem" requires that, in order that y'= f(x,y) have a solution in a neighborhood of (x0,y0), f must be continuous and Lipschitz in that neighborhood. That in itself reduces to measure 0 under any "reasonable" measure.