# Infinite Solution to a differential equation problem

1. May 3, 2012

### philthekinggg

Give an example of an everywhere continuous function F such that the initial value problem y' = F(y), y(0) = 0 has infinitely many solution.

The only function that I have thought of so far that has infinite solutions is y=ey + c, but that obviously doesn't fall under the condition of y(0) = 0.

2. May 3, 2012

### clamtrox

Do you remember the requirement for uniqueness of solutions to differential equations?
If you have a differential equation of the form y'(x) = f(y), where f(y) is continuous, but not Lipschitz continuous, then that might work!

3. May 3, 2012

### HallsofIvy

Staff Emeritus
"Lispschitz" lies part way between "continuous" and "has a continuous derivative" (many textbooks give "has a continuous derivative" as a sufficient but not necessary condition for existence and uniqueness) so if you use a function that is continuous but does not differentiable with respect to y, that will be sufficient.