How to convert the following second order ode to a system of first order odes?

Question

I am given the following ode:

$$x''=mg\sin(\theta)+k(L-\sqrt{x^2+h^2})\left(\frac{x}{\sqrt{x^2+h^2}}\right)-bx'$$ where $x$ is a function of $t.$ How do I convert this equation to a system of first order odes? I tried the usual technique of substituting $x_1=x$ and $x_2=x'$ but it does not work since the ode is non-linear. Perhaps there is another way?

Answer 1

The method is pretty much the same:

$x_1=x$

$x'=x_2$

$x_2'=C_0+k\left(L-\sqrt{x_1^2+h^2}\right)\frac{x_1}{\sqrt{x_1^2+h^2}}-bx_2$.

If you want to get system of linear first order ODE you get into problems, I would try something along the lines of $x_1=x', \, x_2=\sqrt{x^2+h^2}$ but if you use this you have problems with expressing $x_2'$ using only $x_1,x_2$.