Find all solutions of the differential equation
on the interval . Prove that all of these solutions go to 0 as , but that there is exactly one solution which has a finite limit as .
Proof. Since for any we divide both sides by to get
Then, we apply Theorem 8.3 (page 310 of Apostol) with
We choose and compute solutions in terms of . Using partial fractions we evaluate ,
This gives us all solutions to the differential equation in terms of the constant .
Next, we take the limit
for all (since as and all of the other terms in the product are tending to finite limits as well). Next, if
is finite then we must have ; thus,