## Second Order DE: Nonlinear Homogeneous

I am sure most of you are familiar with the equation: m(x)''+c(x)'+k(x) = 0. Then, we create an auxillary equation that looks like this: mr^2+cr+k = 0. And, then we find the roots of this auxillary equation, calling them r1 and r2. And, if the roots are r1,r2>0 we consider the system to be overdamped and we develop the following general form of the equation to be: x=A*e^(r1*t)+B*e^(r2*t) etc...

I have an inverted pendulum with damping and understand the equation to be in the form of a second order nonlinear (homogenous) differential equation.

The second order nonlinear homegenous differential equation is: m(x)''+c(x)'-k*sin(x)=0.
I have tried everything I know and haven't had much luck... I mean how do you find the roots to something like this? I am sure there is a method to this madness. I would think there is some type of variation of parameters or substitution involved but do not know how to apply substitution etc... to this application.

I would appreciate any help or advice that might lead me in the right direction...