# Final condition instead of initial condition

1. Aug 24, 2010

### Petr Mugver

Let's consider a second order differential equation

$$f(x,\dot x,\ddot x,t)=0$$

and let's suppose that f satisfies all the conditions of the Cauchy Theorem, i.e. f is such that the equation above with the initial conditions

$$x(t_0)=x_0\qquad\dot x(t_0)=v_0$$

has an unique solution in a certain neighbourhood of t_0, for every t_0.

My question is, if instead of the two initial conditions above I have an initial and a final condition

$$x(t_0)=x_0\qquad x(t_1)=x_1$$

under what further conditions on f the solution exists for all x_0 and x_1?

2. Aug 25, 2010

### Eynstone

The existence depends crucially on the nature of the equation. The solution is, in general, not unique.

3. Aug 26, 2010

### Petr Mugver

Can you give me some examples? (of a f that satisfies the conditions of my first post but whose solution is not unique for some choice of initial and final conditions)