- Prove that there is exactly one solution of the second-order differential equation
whose graph passes through the points and where , where .
- Is part (a) ever true if for ?
- Generalize part (a) for the second-order differential equation
Include the case .
- Proof. The equation is of the form
Therefore, using Theorem 8.7 (pages 326-327 of Apostol) we have and . Since we have solutions given by
The conditions in the problem tell us and . From the first condition we have
Using this expression for and the second condition we have
where we use that so . Thus, and are uniquely determined, so the solution is unique
- No, if , then and we find the choice of is arbitrary.
- If , then and so the first condition implies
Then, the second condition implies
Thus, is uniquely determined.
If , then implies . Therefore,
Thus, and are uniquely determined as long as .