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 $\displaystyle x'=f(x)$, where x=x(t) is an unknown

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

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

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

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

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

Show all solutions are described by the family

$\displaystyle x(t)={\phi}(t-c)$, where $\displaystyle {\phi}$ 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 $\displaystyle \frac{d}{dx}[f(x)]=\frac{d}{dt}\int_{x_{0}}^{x(t)}\frac{1}{f(s)}ds$

Just a little hung up.