I have an ode of the following form $$y''+by'=a+f(y)$$ where $$a=mg\sin(\theta),\quad f(y)=k y\left(\frac{L-\sqrt{y^2+h^2}}{\sqrt{y^2+h^2}}\right)$$ are constants. How do I find the equilibrium points for this equation?
For an equilibrium solution, I tried to put $y'=0\implies y''=0.$ Hence $$\sin(\theta)=\frac{ky(\sqrt{y^2+h^2}-L)}{mg\sqrt{y^2+h^2}}.$$ What do I do after this?
Possible answer: Put $y'=z$ then the above ODE reduce to following system of equations: \begin{eqnarray} \begin{cases} z'=a-bz+f(y)\\ y'=z. \end{cases} \end{eqnarray} Then the equlibrium solution is: Solution of the system $z'=0$ and $y'=0$ simultaneously.