Thread: Lipschitz Condition-ODE

1. Let f(x,y) be the function:
f(x,y)=x*e^(-|y|) + sin(2x-y)
Prove that there's an M for the function f(x,y) such as:
|f(x,y1)-f(x,y2)| <= M|y1-y2| in every closed rectangle:
D1= { (x,y) | a<= x <=b , -infinity<y<infinity }

2. Given this ODE:
y' = 2|y|sinx
y(x0)=y0

Prove that this ODE has only one soloution...
It's obvious we need to use Lipschitz condition here too, but I'm not sure if the function is defined in all R... This is my only problem in 2...
About 1- I can't prove it...

Help is needed!!

TNX in advance

NVM I've managed to prove it on my own

TNX & sry