# Verify Lipschitz condition

Consider the initial value problem $$\frac{dy}{dx} = x^2 + y^2 \\ y(0) = 0$$

on $[-0.5, 0.5]\times [-0.5, 0.5]$

Find $[-a, a]$ such that the solution exists and is unique so there are two steps involved apparently.

1) verify the Lipschitz continuous condition

2) Fix the parameters M and h

I am able to get $$M = \sup(x^2 + y^2) = 0.25 + 0.25 = 0.5$$ and $$h = \min\{a, \frac{b}{M}\} = \min[0.5, \frac{0.5}{0.5}] = 0.5$$

But the problem I have is with step $1$. I do not know how to verify the Lipschitz condition. If someone can help explain it, that would be awesome.

• You can explicitly solve the differential equation and take it from there. Or is the idea to obtain the Lipschitz condition without doing so? Commented Mar 27, 2014 at 11:49
• Could you please explain how to solve the equation explicitely? Commented Mar 27, 2014 at 11:52

You can find the Lipschitz costant as $$L=\sup\Bigl|\frac{\partial}{\partial y}(x^2+y^2)\Bigr|.$$
• Because the theorem of existence and uniqueness for the equation $y'=f(x,y)$ states that $f(x,y)$ should be continuous (as a function of two variables) an Lipschitz with respect to $y$. Commented Mar 28, 2014 at 15:01