For the first-order system , show that the right-hand side satisfies a Lipschitz condition in the domain for , where and are arbitrary but finite numbers. Deduce that the IVP and has a unique solution for and obtain an estimate for . By allowing and to be as large as possible, attempt to improve your estimate of .
The Lipschitz constant is .
The magnitude of gives an upper bound:
Pick so that .
Improve the estimate for by allowing and to be as large as possible.
The derivative is .
Just set the numerator equal to 0.
Now let so that