Can a problem formulated as an initial value problem be converted to a final value problem and vice-versa?
I don't really have an example for this. But, I would like to know is it possible, if so how to do it, what are all the situations that possess this kind of characteristics and if not, why can't we do it?
Thanks for your answers.
If you are given a differential equation $$y'=f(t,y)\qquad\bigl((t,y)\in\Omega\bigr)$$ defined in some domain $\Omega\subset{\mathbb R}^2$, as well as a point $(t_0,y_0)\in\Omega$, you can always look at the solution $t\mapsto \phi(t)$ satisfying $\phi(t_0)=t_0$ not only for $t\geq t_0$, but also for $t\leq t_0$. In other words: The function $\phi$ is also the solution that "terminates" at $(t_0,y_0)$.
In other words: If you are given the closing time $T$ and somehow want to realize $\phi(T)=y_T$ for given $y_T$ then finding the value $y_0$ at time $t_0$ producing the given end value $y_T$ amounts to completely solving the "terminating value problem" described in the first paragraph corresponding to the "terminal data" $(T,y_T)$.
