Final condition instead of initial condition

[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?

 

[Eynstone]

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

 

[Petr Mugver]

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

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)