# Stability Region of ODE Numerical Method (Runge-Kutta)

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.

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.