If I have a set of differential equations describing the system in a regime, of the form:
$\ddot x= a(t) \dot x + by+cx \\ \dot y=d(t)\dot x+ey+f(t)$
Now this system behaves according to this: $\dot x=g(t) + h(t)x$
in a different regime. I am supposed to vary a parameter to reach the two limiting cases. But I want to know the behavior in the middle. Is there a way to smoothly vary the governing equations so that I can then solve it with an Euler like method? If I just vary the parameter, then the first set of equations become unstable. Of course it does, since I am to remove the second order term.
Edit: What I am asking is that if we have a set of coupled ODEs, and we want to vary some parameters in them. From physical point of view, say I want to vary friction coefficient $\gamma$ such that we can rewrite the equation in a way that the highest order derivative disappears. Now analytically we can now write everything nicely, but numerically, in Euler method, we put the highest order derivative in the LHS. If that term itself is to disappear by taking the limit, how to avoid instability of the solution? I want to smoothly vary the parameter and yet obtain stable solutions. Is that possible?