Hi,

I'm have some trouble understanding what my lecturer did.

The question is:

For the IVP, $\displaystyle y' = \frac{x+y}{2y}$, $\displaystyle 0 \le x \le 1$, y(0)=1,

find an appropriate Lipschitz constant L in the domain $\displaystyle \{(x,y) : 0\le x \le 1, y \ge 1\}$. Hence determine whether or not the soln y(x) is unique.

We've found L = 1/2 (by finding $\displaystyle f_y$ and maximizing in our domain, but doesn't this function have a singularity at y=0? Which means the uniqueness thm doesn't apply?

Can someone explain?

EDIT: Also, he did things like:

y(0)=1 => (x+y)/2y > 0 => y' > 0

=> y increases

=> y >= 1

Which I don't understand.