- #1
Petr Mugver
- 279
- 0
Let's consider a second order differential equation
[tex]f(x,\dot x,\ddot x,t)=0[/tex]
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
[tex]x(t_0)=x_0\qquad\dot x(t_0)=v_0[/tex]
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
[tex]x(t_0)=x_0\qquad x(t_1)=x_1[/tex]
under what further conditions on f the solution exists for all x_0 and x_1?
[tex]f(x,\dot x,\ddot x,t)=0[/tex]
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
[tex]x(t_0)=x_0\qquad\dot x(t_0)=v_0[/tex]
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
[tex]x(t_0)=x_0\qquad x(t_1)=x_1[/tex]
under what further conditions on f the solution exists for all x_0 and x_1?