A dif equation, proving periodic solutions

[Aerosion]

Homework Statement

Based only on differential equations, discuss why it is unlikely that d^2y/dt = ay with a > 0 has periodic solutions.

Homework Equations

periodic functions go to infiniti or negative infiniti, differential equations are continuous...

The Attempt at a Solution

so I said that the equation should be concave. the equation should be differentialble everywhere and has to be continuous with infinitely many inflection points. my teacher said that I'm on the right track but i need help putting my proof into words. any help?

 

[mighty2000]

Dear fellow scientist,

Thank you for bringing up this interesting topic. Let us consider the differential equation d^2y/dt = ay, where a is a positive constant. This equation can be rewritten as d^2y/dt^2 = a^2y.

First, let us define what we mean by a periodic solution. A periodic solution is a function that repeats itself after a certain period of time. In other words, the function has a constant period, meaning that the value of the function at any given time t is equal to the value of the function at time t+T, where T is the period.

Now, let's think about the behavior of the function y in this differential equation. Since the second derivative of y is directly proportional to y, the function y must have a concave shape. This means that the function y must have infinitely many inflection points, as you correctly pointed out.

Furthermore, we know that periodic functions have a limit as they approach infinity or negative infinity. However, the function y in this differential equation has no limit as t approaches infinity or negative infinity. This is because the function y grows exponentially with time, since the second derivative is directly proportional to the function itself.

Therefore, we can conclude that it is unlikely for this differential equation to have periodic solutions, as the function y does not have a constant period and does not have a limit as t approaches infinity or negative infinity.

I hope this helps to further explain why it is unlikely for this differential equation to have periodic solutions. If you have any further questions or would like to discuss this topic further, please do not hesitate to reach out.