In particular, what could be a benefit of linearizing a system of first order non-linear ODEs? I have seen a number of papers derive linearized versions of non-linear systems of ODEs, but I can't seem to find much of an argument as to why those are better than the original non-linear system.
For example, given a system of 3 ODEs $\dot x=f_x(x,y,z) $ , $\ \dot y=f_y(x,y,z)$ , $\ \dot z=f_z(x,y,z)$, say you derive some linear system $\dot x= a_1(t)x+a_2(t)y+a_3(t)z$ , $\dot y= b_1(t)x+b_2(t)y+b_3(t)z$ , $\dot z= c_1(t)x+c_2(t)y+c_3(t)z$, and that this linear system provides a decent approximation to the nonlinear system.
To me, from a couple weeks of searching the internet, it seems there aren't really any benefits unless the linear system has certain properties (such as constant coefficients, as is the case with the Clohessy Wiltshire equations) that let you solve for some explicit solution. It seems to me that, without an explicit solution, all you have is another set of differential equations to plug into an integrator. At that point you might as well just use the original nonlinear system (i.e. use the integrator to approximate the solution to the original equation rather than an approximation of the original equation).
I was wondering if there are any advantages other than the possibility of an explicit solution. For example, could a linear system somehow improve computational efficiency? I have been browsing the internet for a little while now, but it has proven hard to sift through material about linearizing systems and solving linear ODEs etc.