tough DE theory problem

**galactus** · September 12th 2008, 10:56 AM

Hey Gang:

I am going to ask a question this time. I got a hold of the book

Ordinary DE and Stability Theory by Sanchez I picked up for a song. It has some tough problems. I thought it would be a nice book, but the notation is killer.

I have been wanting to strengthen my DE skills.

Here is one if anyone has a good idea.

Let's take the equation $x'=f(x)$ , where x=x(t) is an unknown

scalar function and f and df/dx are defined and continuous on the strip

$B=[(t,x)|-\infty<t<{\infty}, \;\ a<x<b]$ .

We can also assume $f(x) \neq 0, \;\ a<x<b$ ,

and let $F(x)=\int_{x_{0}}^{x}\frac{1}{f(s)}ds$ ,

where $a<x_{0}<b$ .

Show all solutions are described by the family

$x(t)={\phi}(t-c)$ , where ${\phi}$ is the inverse

of F, and c is a constant determined by initial conditions.

I tried using the second fundamental theorem of calculus and thought

about $\frac{d}{dx}[f(x)]=\frac{d}{dt}\int_{x_{0}}^{x(t)}\frac{1}{f(s)}ds$

Just a little hung up.

**ThePerfectHacker** · September 12th 2008, 12:56 PM

Wingless sent me the following. He gets the credit for this post

We know these:

$\frac{dx}{dt} = x'(t) = f[x(t)]$

$F[x(t)] = \int_{x_0}^{x} \frac{1}{f(s)}~ds$

Now, on the RHS of the second line, substitute $s = x(t)$ to get,

$F[x(t)] = \int_{x^{-1}(x_0)}^{x^{-1}[x(t)]} \frac{x'(t)}{f[x(t])}~dt$

Using the first line, replace x'(t) to get,

$F[x(t)] = \int_{x^{-1}(x_0)}^{x^{-1}[x(t)]} \frac{f[x(t)]}{f[x(t])}~dt = \int_{x^{-1}(x_0)}^{x^{-1}[x(t)]}dt = \underbrace{x^{-1}[x(t)]}_{\text{this is t}}-\underbrace{x^{-1}(x_0)}_{\text{some constant}}+C = t- C$

$x(t) = \phi (t-C)$

**galactus** · September 12th 2008, 01:40 PM

Thank you much fellas. Since the wing isn't here, tell him thanks as well.

I started out right and then went down the primrose path

**bkarpuz** · September 12th 2008, 02:55 PM

My solution is also similar to that by wingless's, however I would like to give it here for showing some technical details.
Since $x^{\prime}(t)=f(x(t))\neq0$ for all $x\in(a,b)$ , we see that $x$ is either increasing or decreasing, i.e. it has a inverse function $x^{-1}$ .
Now, consider the followings.
$F[x(t)]=\int\limits_{x_{0}}^{x(t)}\frac{1}{f(s)}ds=\int\l imits_{x_{0}}^{x(t)}\frac{1}{x^{\prime}(x^{-1}(s))}ds=\int\limits_{x^{-1}(x_{0})}^{t}dv=t-x^{-1}(x_{0}),\qquad(\ast)$
where
$v=x^{-1}(s)\text{ and so }dv=(x^{-1}(s))^{\prime}ds=\frac{1}{x^{\prime}(x^{-1}(s))}ds$ (see Inverse functions and differentiation - Wikipedia).
Applying $\phi$ on both sides of (*), we get the desired result.

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