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).

 

[tacman]

Yes, it is possible to determine if a solution to a differential equation is periodic without knowing the explicit solution. This can be done by analyzing the behavior of the solution over time. If the solution repeats itself after a certain period of time, then it is periodic. This can be seen by graphing the solution or by using mathematical techniques such as Fourier analysis.

It is also possible to generate differential equations that have periodic solutions. In fact, many real-world phenomena can be modeled using periodic differential equations. Some examples include pendulums, electrical circuits, and population growth models.

In the case of y" = y(y-10)(50-y), the behavior of the solution will depend on the initial conditions and the parameters of the equation. Depending on the values of these parameters, the solution may or may not be periodic. In general, it is not possible to determine the periodicity of a solution without knowing the specific values of the parameters. However, there may be certain values of the parameters that guarantee periodicity, such as in the case of y" = -ay.

In conclusion, while there may not be a general analytical way to determine the periodicity of a solution to a differential equation, it is possible to analyze the behavior of the solution to determine if it is periodic. Additionally, it is possible to generate differential equations that have periodic solutions, making them a useful tool for modeling real-world phenomena.