Consider the following coupled system of differential equations:
Our unknowns are the functions $u_1(t)$, $u_2(t)$ and $u_{GV}(t)$
Let's say $k_{GV}$ is a nonlinear function of the form $k_3=f(u_{GV})$.
$F_1$ and $F_{GV}$ are functions of time imagine sine waves for example. All the other terms appearing in the system are constants.
Now lets say I solve the system and obtain all the results $u_1$, $u_2$ and $u_{GV}$.
I decide then to consider the following problem:
$m_{GV}\ddot{u}_{GV}(t)-k_3u_2(t)+(k_3+k_{GV})u_{GV}(t)=F_{GV}$
In this problem I consider that the terms $\ddot{u}_{GV}$ and ${u}_{GV}$ are unknowns but I consider the same $u_2(t)$ that we obtained earlier. I solve again for ${u}_{GV}$.
Logically shouldn't the new function ${u}_{GV}$ that I obtain be different from the one I obtained from the full system earlier?
I did some simulations on Mathematica and used different $k_{GV}$ some of which are very non linear but I always obtained the same solution. Does this make sense? Is there something I'm not understanding well?
Thank you