Determine if a solution to a differential equation

[JoAuSc]

Is it possible to determine if a solution to a differential equation is or isn't periodic, even if you don't know the solution explicitly? Also, is it possible to generate differential equations that have periodic solutions (besides the obvious ones like the solution to y" = -ay)? The reason why I'm asking is that I was fooling around with graphing y" = y(y-10)(50-y) in Maple (for y(0) = 20, y'(0)=0), but depending on what the parameters of the graph are I seem to get different answers. I'm just wondering if there's an analytical way to determine periodicity.

 

[HallsofIvy]

To give a simple, special case of your question: the solutions to a linear differential equation with constant coefficients are periodic if and only if the characteristic values are imaginary.

For any a, ai is pure imaginary and has conjugate -ai. (r-ai)(r+ai)= r²+a²= 0 has those as as solutions and so the differential equation y"+ a²y= 0 has solutions cos(ax) and sin(ax). If you want to "fancy" it up a bit more add some other factors say (x- b)(x²+a²= 0 is the characteristic equation for a d.e. that has solutions e^bx as well as sin(ax) and cos(ax).