Lagrange's equation of motion is:
$\begin{align*}\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot q} = 0\end{align*}$
Plugging in the given expression for $L$ , we find that
$\begin{align*}4 b q^3 - 2 a \ddot q = 0 \Rightarrow \boxed{\ddot q = \frac{2 b}{a} q^3}\end{align*}$
Therefore, answer (D) is correct.

Solution to 2001 Problem 74