I have vector ODE like this.
$$y''=y+y'$$ I want to solve this numerically.

To solve explicitly,
iterarion formula is like this.
However, due to coefficient of my equation,
this iteration will be unstable.

So I want to use implicit method.
How to construct implicit scheme for this?


Why don't you use Runge-Kutta for that equation? Since we have more than one method to solve such ODE. For instance, we can put $u=y'$, which lead us to the system of ODEs $$ \begin{array}{lcl} y' &=& u \\ u' &=& u + y \end{array} $$ Or one has $$ W' = f(W,t) $$ where $W(t)=( y(t), u(t) )$, $f =(f_1,f_2)$ and $$ f_1(W,t) = u(t),~~~ f_2(W,t) = u(t) + y(t). $$ Then we can solve $W'=f(W,t)$ by using Runge-Kutta method. (You also need an initial condition, e.g. $y(0)=y_0$ and $y'(0) = y_1$, or in vectorform we have $W(0) = (y_0, y_1)$ ).

Edit: From the derived vector equation $W'=f(W,t)$ we can also use implicit schemes for time-advance.


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