Hello everybody, thank you very much for reading!
I need to solve the following problem (I'll try my best in English, it's not my native tounge):
y' = y^*(1/3) + 1, y(0)=0, and let f(x,y) = y^(1/3) + 1
f(x,y) doesn't satisfy Lipschitz's condition on y=0. (that's a given).
The question is:
Does the problem has one solution or not? Then they add: What is the conclusion on Lipschitz's condition in the existence and uniqueness theorem?
1) I know how to prove an equation such as this has one and only solution, but I don't know how to prove an equation has more than one... I mean, there are several conditions for a function to satisfy so that this type of problem will have one solution, but if it doesn't satisfy them - does that mean there isn't one solution?
How do I approach this problem?
2) let D be the area in which f(x,y) is continuous. I know that if f(x,y) satisfies Lipschitz's condition in every partial area to D, then the existence and uniqueness law is being satisfied. I know nothing more than that. What are they implying to?
Thank you very much! I hope I was clear enough...