# When does an initial value problem have exactly two solutions.

My book have examples of an initial value problem which have unique solution and some which have infinite solutions.But I want to know when an initial value problem have two solutions. Is possible or not. Thanks in advance

• You can give any type of example of an initial value problem with exactly two solutions. – Kulwinder Singh Dec 10 '15 at 4:46

The generic situation is that if there is no uniqueness of solutions, then there are infinitely may solutions. Under reasonable conditions on the equation $y'=f(x,y)$, $y(x_0)=y_0$, it can be shown that there is a unique solution or there exist a maximal solution $\phi(x)$ and a minimal solution $\psi(x)$ such that $\phi(x_0)=\psi(x_0)=y_0$ and $\phi(x)<\psi(x)$ for $x\ne x_0$ in the interval of existence. In the last case, the region $\{(x,y):\psi(x)<y<\phi(x)\}$ is covered by the graphs of infinitely many solutions of the initial value problem. This is the situation for the well known example $y'=\sqrt{y}$, $y(0)=0$.