# Smallest interval for existence of unique solution of differential equation

#### shashank dwivedi

The smallest interval on which a unique solution exist for the IVP yâ€²=e^(2y), y(0)=0, is what?

A)|x|<=1/2e
B)|x|<=2e
C)|x|<=2/e
D)|x|<=1/e

#### Greens

Page 6 has the needed theorem.

Since $f=y'=e^{2y}$ and $\frac{\partial f}{\partial y}=2e^{2y}$ are continuous everywhere in $\mathbb{R}$, any interval of $x$ containing $(0,0)$ will have a unique solution satisfying the IVP.

In that case we should just be able to choose the smallest interval, since all of them work. $|x| \leq \frac{1}{2e}$ in this case.

1 person

#### skipjack

Forum Staff
In that case, what would the unique solution be for the cases (B) and (C)?