# variation of a final state due to changes in period (where the period is a parameter)

I have a simple ordinary differential equation

$\frac{dx}{dt}=f(x,t,p,T)$

$x(0) = x_0$, $x(T) = x_T$

where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks!

NOTE: I know I can compute $\frac{dx_T}{dp}$ integrating the equation $\frac{d}{dt} (\frac{dx}{dp}) = \frac{df}{dx}\frac{dx}{dp}+\frac{df}{dp}$ from $t=0$ to $t=T$.

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This makes no sense to me -- if $x_T$ is not a given boundary condition, then what is it being determined by? Also, you say $T$ is a constant parameter and then you form a derivative with respect to $T$ -- then in what sense is $T$ constant? –  joriki Oct 31 '12 at 9:21
Hi joriki. $x_T$ is not given - I'm only defining it as the final state at time T; T is constant meaning is not a function of x nor t. You can see it as a problem parameter, see the example with $p$. Thanks –  user47662 Oct 31 '12 at 10:37