I have the following system of second order differential equation :
$\left\{ \begin{split} &\frac{dy}{dt}=F(t,x,y) \\ &\frac{dx}{dt}=y \end{split} \right.$
with $y(t)$ and $x(t)$ defined in the interval : $I=[0,1]$. I would have liked just to compute : $\left\{ \begin{split} &y_{k+1}=y_k+ F(t_k,x_k,y_k)\Delta t \\ &x_{k+1}=x_k + y_k \Delta t \end{split} \right.$
But I have two boundary conditions defined at different points, so that I have no point to start with :
$\left\{ \begin{split} &y(0)=0 \\ &x(1)=x_0 \end{split} \right.$
How could I proceed, with that method or another one ?