# ODEUET1

From Example Problems

Consider the initial value problem

where is a positive integer. Find all values of for which there is a unique solution. In the latter case, justify your answer by using an appropriate theorem.

Let .

Check the Lipschitz condition.

So is Lipschitz if is bounded.

Solving the DE,

There is no solution to this DE.

Let .

Check Lipschitz.

is true.

Solve the DE.

if ,

is the unique solution.

If ,

is the unique solution.

If ,

is the unique solution.

Let .

Test Lipschitz.

But is not bounded by a constant so is not Lipschitz and therefore there is not a unique solution.

Solve the DE.

So

There are infinitely many solutions because the equation holds for all .