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