Given the ODE $$ \ddot{X} + \frac{3}{t}\dot{X} + F(X) = 0,$$ what would be a stable explizit discretization?
One can rewrite the ODE as a first order equation of the form $$ \dot{Y}_1 = Y_2 \\ \dot{Y}_2 =-\frac{3}{t}Y_2-F(Y_1), $$ by setting $Y_1:=X, Y_2:=\dot{X}$. Then one can apply a Runge-Kutta-method or just a simple Euler, but how do I check that the method should be stable since the ODE considered here is not autonomous.
EDIT: You are asking about stability, then this may help you:
Under certain assumptions you may be able to prove that your two dimensional system is locally one-sided Lipschitz continuous (depending on where you seek a solution, and what $F$ is, looks like you need a Lipschitz constant of F being > 1), and then use a B-stable RK method (B-stable in the sense of Butcher, ambiguous term). Then I need more information. B-stability is briefly described on the Wikipedia page of RK methods.