Consider the Riccati equation

This equation has two constant solutions. Starting with these and the previous exercise (linked above) find further solutions in the cases:

- If , find a solution on the interval with when .
- If or , find a solution on the interval with when .

First, we find the two constant solutions. If is constant then so

From Exercise 19 (linked above) we know we can obtain additional solutions to the Riccati equation by

where is a solution of

In the present case we have , so is the solution of either

for the cases and , respectively. Each of these is an first-order linear differential equation which we can solve using Theorem 8.3 (page 310 of Apostol). For the first one we have , and let , and . Then we have

This gives us the first solution

Evaluating the second differential equation, this time with , , and we have,

Therefore, the second solution is

Finally, for the specific cases in (a) and (b).

- We want , so we choose . Then . (This follows since .) Therefore, , so,
where .

- In this case we want or , so we choose . Since we have . Therefore,
where .