Consider the second-order differential equation
If is a point in the plane and if is a given real number, prove that this differential equation has exactly one solution whose graph passes through the point and has slope . Consider also the case .
Proof. If , then implies , and implies . This gives us . Then, so
If , then so . This means the solutions are of the form
The condition then gives us
The condition give us
Thus, we have a unique solution with and as given above