Equation of motion Pendulum using $w=e^{ix}$ I'm working with the equation of motion for a pendulum as follows:
$$x''+ \frac{g}{l} \sin (x)=0$$
Where $x$ is the angle between the pendulum and the vertical rest position.
I am required to use the complex variable $w=e^{ix}$ to rewrite the equation of motion in the form $(w')^2= Q (w)$, where $Q$ is a cubic polynomial. So in the form $(u')^2=u^3 + au + b$, with $a$, $b$ constants.
I'm not sure where to start with the question, can anybody help me get going?
Homework help

Multiply the equation through by $x'$ and integrate once to get
$$x'^2-\frac{2 g}{\ell} \cos{x} = C$$
where $C$ is a constant of integration.  Now, if $w=e^{i x}$, then $\cos{x}=(w+w^{-1})/2$ and 
$$w' = i x' e^{i x} \implies x'=-i w'/w$$
Then the equation is equivalent to
$$-\frac{w'^2}{w^2} - \frac{g}{\ell} \left (w+\frac{1}{w}\right)=C$$
Then, multiplying through by $-w^2$, we get
$$w'^2+\frac{g}{\ell} w^3 + C w^2+\frac{g}{\ell} w=0$$
which is not quite the form specified, but is an equation of the form $w'^2+Q(w)=0$, where $Q$ is a cubic in $w$.
