# Is the interval I of an autonomous diff closed?

Is the interval I of an autonomous diff closed???

## Homework Statement

Given this autonomous diff.eqn

Where we have an open set E defined on R^n and $$f \in \mathcal{C}^1(E)$$

$$x' = f(x)$$ where $$x(t_a) = x(t_b)$$ and $$t_a,t_b \in I$$ and where $$t_a < t_b$$.

Show for n = 1, that the solution x is constant.

## The Attempt at a Solution

According to the definition of the autonomous differential equation. Then for x to be a solution to the diff.eqn then certain conditions must upheld.

1) x(t) is differentiable on I and $$\forall t \in I$$, $$x(t) \in E$$ and

2) $$x'(t) = f(x(t))$$.

according to how f is defined its continous on E and have first derivatives on E. E is open on subset on $$\mathbb{R}^n$$ and in our case it must mean that if E is defined as $$E \subset \mathbb{R} \times \mathbb{R}$$ and then $$I \in \mathbb{R}$$ which is the interval of all solutions of the original problem. Then if $$x(t_a)$$ and $$x(t_b)$$ are both solution of the original equation. Then it must mean (as I see it!) that x(t_a) = x(t_b) = 0. Hence a t_c defined on I will result in $$x'(t_c) = f(x(t_c)) = 0$$. Thusly for n = 1 x is a constant solution of of the autonomous diff.eqn x' = f(x).

maybe I have misunderstood something but isn't if E is a subset of R x R and I is subset R doesn't mean that both sets E and I are subset of the same set R?

Have expressed this satisfactory?

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Dick
Homework Helper

i) E isn't a subset of RxR. It's a subset of R. You are given it's a subset of R^n and n=1. ii) f(t_a) isn't a solution to the DE. It's not even a function of t. It's a single point on the solution. I think the point here is that if n=1 then the DE is separable, You can express t as a function of x.

i) E isn't a subset of RxR. It's a subset of R. You are given it's a subset of R^n and n=1. ii) f(t_a) isn't a solution to the DE. It's not even a function of t. It's a single point on the solution. I think the point here is that if n=1 then the DE is separable, You can express t as a function of x.

Hi Dick,

You mean f(t_a) and f(t_b) are points on I ?

and also I consider that that interval I and the set E are both subsets of R? Aren't they?

Don't I need to show here that since we know has first derivatives on E (according to my original post) and is continious on E. Then since E is a subset of R for n = 1.

Then for x to be a solution of f then it has to be differentiable on I?

Dick
Homework Helper

Hi Dick,

You mean f(t_a) and f(t_b) are points on I ?

and also I consider that that interval I and the set E are both subsets of R? Aren't they?

Don't I need to show here that since we know has first derivatives on E (according to my original post) and is continious on E. Then since E is a subset of R for n = 1.

Then for x to be a solution of f then it has to be differentiable on I?

Hi, Susanne. t_a and t_b are points on I. f(t_a) and f(t_b) are points in the subset E. Yes, they both subsets of R. And yes, x(t) is differentiable on I (as a function of t) and f(x) is differentiable on E as a function of x. But they don't have all that much to do with each other. One is the domain of x and the other is the domain of f and contains the range of x. Look. Solve x'=x (where '=d/dt) on the interval t_a=1 and t_b=2. That is the sort of problem they are talking about.

Hi, Susanne. t_a and t_b are points on I. f(t_a) and f(t_b) are points in the subset E. Yes, they both subsets of R. And yes, x(t) is differentiable on I (as a function of t) and f(x) is differentiable on E as a function of x. But they don't have all that much to do with each other. One is the domain of x and the other is the domain of f and contains the range of x. Look. Solve x'=x (where '=d/dt) on the interval t_a=1 and t_b=2. That is the sort of problem they are talking about.

okay thanks :)

is x(t_a) and x(t_b) are points on the subset I?

I have come up with my own idear using Rolle's theorem. So please bare with me :)

Let E defined as in the original post. The solution x is defined on I. Therefore the diff.eqn is differentiable on I. Assume that $$x(t_a), x(t_b) \in E$$. Then by Rolle's theorem the diff.eqn has both maximum and minimal solution on I. Thusly none of these are obtained from an interior point of I. Thusly $$x(t_a)= x(t_b)$$ and x is there constant on I.

Because I need to show the above problem with Rolle's theorem in mind.

How is this Dick?

Dick
Homework Helper

I already told you x(t_a) and x(t_b) aren't in I. They're in E. And my version of Rolle's theorem doesn't say anything about maxima or minima. The rest of what you are saying makes even less sense. Are you using theorems I don't know about? How do you know the extrema aren't in the interior?

But it is true that if x has a interior maximum on [t_a,t_b] you can see the DE can't be autonomous. Loosely speaking, you can draw a horizontal line a little below the maximum, say at t_max and find t+<t_max<t- such that x(t+)=x(t-)<x(t_max) but x'(t+)>0 (since it's heading up to the max) and x'(t-)<0 since it's heading down from the max. I'm not sure how to make that completely rigorous just now. But do you see why that would lead to a contradiction?

I already told you x(t_a) and x(t_b) aren't in I. They're in E. And my version of Rolle's theorem doesn't say anything about maxima or minima. The rest of what you are saying makes even less sense. Are you using theorems I don't know about? How do you know the extrema aren't in the interior?

But it is true that if x has a interior maximum on [t_a,t_b] you can see the DE can't be autonomous. Loosely speaking, you can draw a horizontal line a little below the maximum, say at t_max and find t+<t_max<t- such that x(t+)=x(t-)<x(t_max) but x'(t+)>0 (since it's heading up to the max) and x'(t-)<0 since it's heading down from the max. I'm not sure how to make that completely rigorous just now. But do you see why that would lead to a contradiction?

I don't Dick,

They use Rolle's theorem to show bla bla. But then they don't say. Which part of Rolle's theorem? The whole part or just the idear of it :(

Dick
Homework Helper

I don't Dick,

They use Rolle's theorem to show bla bla. But then they don't say. Which part of Rolle's theorem? The whole part or just the idear of it :(

Who is 'they'? Are you trying to present the argument given as a solution? If so can you give me the whole thing word for word? Would you also state what you think Rolle's theorem is?

Who is 'they'? Are you trying to present the argument given as a solution? If so can you give me the whole thing word for word? Would you also state what you think Rolle's theorem is?

Hi Dick,

The original post is in post number 2 in this thread. "They" are the people who formulated the problem ;)

Rolle's theorem which I'm trying to use comes from Apostol's Mathematical Analysis p. 110.

I know this book pre-dates the Vietnam war but it was the one we used on our Mathematical Analysis course.

Dick
Homework Helper

Hi Dick,

The original post is in post number 2 in this thread. "They" are the people who formulated the problem ;)

Rolle's theorem which I'm trying to use comes from Apostol's Mathematical Analysis p. 110.

I know this book pre-dates the Vietnam war but it was the one we used on our Mathematical Analysis course.

Ok, so Rolle's theorem says if x(t_a)=x(t_b) then there is a point t_c in between where x'(t_c)=0. How does this help?