Prove there exists a unique function which is continuous on and that satisfies the equation

for all positive and find the function.

*Proof.* Let

Then we are looking for solutions to the differential equation

We apply Theorem 8.3 (page 310 of Apostol) with

This gives us

Therefore,

Therefore,

By the uniqueness property of Theorem 8.3, we know this solution is unique

Isn’t b=1 for a=1?

No. It is 0