I need to write
$$\frac{d^2\theta}{dt^2} + 4\sqrt{k}\,\frac{d\theta}{dt}+g\sin(\theta)=0$$
as a first order equation.
What I have done so far is:
Let $z = \frac{d\theta}{dt}$ Then $z' = \frac{d^2\theta}{dt^2}$
Therefore the second order equation can be written as
$$z' + z +g\sin(\theta) = 0)$$
But I'm not sure if this is correct.
Let $a=4\sqrt k$ $$\frac{d^2\theta}{dt^2} + a\frac{d\theta}{dt}+g\sin(\theta)=0$$ Change of function : $\frac{d\theta}{dt}=z(\theta)$
Do not confuse with $\frac{d\theta}{dt}=z(t)$ which fails to reduce the order of the ODE.
$\frac{d^2\theta}{dt^2}=\frac{dz}{dt}=\frac{dz}{d\theta}\frac{d\theta}{dt}=z\frac{dz}{d\theta}$
$$z\frac{dz}{d\theta} + a\:z+g\sin(\theta)=0$$ Let $z=\frac{1}{y(\theta)}$ $$\frac{dy}{d\theta}=a\:y^2+g\sin(\theta)\:y^3$$ This is an Abel's differential equation of the "non-solvable" kind, in the sens of "with a finite number of elementary and/or standard special functions".
That means : There is no hope to find a closed form for the solutions.
Numerical methods and/or judicious physical approximation have to be considered to further progress.
