Periodic Solution to FIrst Order ODE Proof

[Opus_723]

Homework Statement

Consider the differential equation x' = f(t,x) where f(t,x) is continuously differentiable in t and x. Suppose that

f(t+T,x) = f(t,x) for all t

Suppose there are constants p, q such that

f(t,p) > 0, f(t,q) < 0 for all t.

Prove that there is a periodic solution x(t) for this equation with p < x(0) < q.

The Attempt at a Solution

Not really sure what approach I'm supposed to take. I imagine I'm supposed to use the fact that f(t,x) is continously differentiable, but I'm not sure what that's supposed to give me. I wrote a few expressions using the fundamental theorem of Calculus, but that looked like a dead end. I'm not sure what I should copy here as previous work since I just tried a bunch of things that didn't seem to lead anywhere. If I could get a hint on what approach to use, that would be great.

 

[mighty2000]

Thank you for posting this interesting problem. I would approach this problem by first understanding the given conditions and what they mean in terms of the differential equation.

The condition f(t+T,x) = f(t,x) for all t means that the function f(t,x) is periodic with a period of T. This means that the value of f(t,x) at any time t is the same as its value at t+T.

Next, the given constants p and q represent two points on the x-axis where the function f(t,x) takes on positive and negative values respectively. This means that the function f(t,x) crosses the x-axis at these points.

Now, let's consider the differential equation x' = f(t,x). This means that the derivative of x with respect to time, or x', is equal to the function f(t,x). This can also be interpreted as the slope of the curve x(t) at any given point (t,x).

From the given conditions, we know that f(t,p) > 0 and f(t,q) < 0 for all t. This means that at the points p and q, the slope of the curve x(t) is positive and negative respectively.

Now, imagine starting at a point on the x-axis between p and q, let's say x(0). Since the function f(t,x) is continuous and differentiable, we can use the Intermediate Value Theorem to show that there must be a point c between p and q where f(t,c) = 0. This means that the slope of the curve x(t) at this point c is 0, indicating a horizontal tangent.

Since the function f(t,x) is periodic with a period of T, this point c will repeat itself at every interval of T. This means that the curve x(t) will have a horizontal tangent at points c, c+T, c+2T, and so on.

Therefore, the curve x(t) will oscillate between the points p and q, creating a periodic solution with p < x(0) < q.

I hope this explanation helps you understand the approach to solving this problem. Let me know if you have any further questions or if you need more clarification.