Stability Region of ODE Numerical Method (Runge-Kutta)

Question

I am self-studying numerical methods solving ODEs, and I came across the following question:

Consider the 4th order Runge Kutta method with $\Delta t = 1$ applied to $\frac{dx}{dt}=i\omega x$. Find the range of $\omega$ for which the method is stable.

I have thought about it for quite a while but still have no idea how to do it. My biggest confusion is that: why does the stability region of a Runge-Kutta method or a linear multi-step method depend on the particular ODE that I'm trying to solve? I learned about the characteristic polynomial, and I know that as long as the zeros of the polynomial $\lambda$ satisfy $|\lambda| \leq 1$ and those $\lambda$ on the unit circle are simple roots, then it's enough. I don't understand why a particular ODE matters.

Thanks so much for everyone's help. Much appreciated.

Answer 1

A step of fourth order Runge-Kutta applied to this differential equation, starting from $x_0$ at $t=0$, should give you $x_1 = c x_0$ for some complex number $c$ (depending on $\omega$). If $|c| \le 1$ the method is stable, if $|c| > 1$ it is unstable.