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?

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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)

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