show function satisfies Lipschitz condition on a domain

Question

I have been given the Initial Value problem

$\frac{dy}{dx} = y^2$, $y(1)=-1$

Let $f(x,y)=y^2$.

I have defined the domain R: $|x-1|\leq a$, $|y+1|\leq b$ where $a,b>0$

I am trying to show that Lipschitz condition is satisfied on R but stuck on trying to finding a value of K. I am stuck after the following steps:

$|f(x,y_1)-f(x,y_2)|=|y_1^2-y_2^2|=|y_1+y_2||y_1-y_2|$

What value of K can I take so that $|f(x,y_1)-f(x,y_2)|\leq K|y_1-y_2|$ and hence satisfying Lipschitz condition on R?

Answer 1

In general, the Lipschitz constant can be found by taking the maximum absolute value of the derivative (this follows from the mean value theorem) over the domain you are looking at. In your case, this means you would only need to find a bound on $|2y|$ for $y \in [-b-1, b-1]$ which should be straightforward enough.

This also coincides with a bound that you could find on $|y_1 + y_2|$, as you can easily check.