Wingless sent me the following. He gets the credit for this post
We know these:
Now, on the RHS of the second line, substitute to get,
Using the first line, replace x'(t) to get,
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 , where x=x(t) is an unknown
scalar function and f and df/dx are defined and continuous on the strip
.
We can also assume ,
and let ,
where .
Show all solutions are described by the family
, where 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
Just a little hung up.
My solution is also similar to that by wingless's, however I would like to give it here for showing some technical details.
Since for all , we see that is either increasing or decreasing, i.e. it has a inverse function .
Now, consider the followings.
where
(see Inverse functions and differentiation - Wikipedia).
Applying on both sides of (*), we get the desired result.