Is the interval I of an autonomous diff closed?

[Susanne217]

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 continuous 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?

 

[Dick]

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.

 

[Susanne217]

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]

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.

 

[Susanne217]

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]

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?

 

[Susanne217]

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]

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?

 

[Susanne217]

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]

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?