I am trying to study the Weierstrass $\wp$ function using a combination of texts from Alfors; Cartan; Freitag and Busam; and Siegel. But I am having some trouble because I would like to try to avoid starting from the Ansatz equation approach given in most of the texts.
I would like to derive the coefficients of $\wp$'s Laurent expansion by considering the differential equation
$$(\wp^{\prime})^{2}=4\wp^{3} -g_{2}\wp(w) - g_{3}.\qquad (1)$$
By writing $$\wp = \sum_{k=-N}^{\infty}a_{k}\omega^{k}\qquad (2)$$
and substituting (2) into (1) to get the form
$$\wp(\omega) = \frac{1}{\omega^{2}} + a_{2}\omega^{2} + a_{4}\omega^{4} + \dots\qquad (3)$$
where I found $(3)$ in Freitag and Busam's Complex Analysis. Intuition I gained from studying the Ansatz equation approach suggests this must be true, but even considering this for small $N$, the combination of the derivative (I am assuming I can safely differentiate term wise) and the cubic are creating multinomial expansions that seem to have no hope of cancellation when powers of $\omega$ are compared on the left and the right hand side.
Using achille hue's suggestion and some additional work, I was able to reduce the problem down to
$$\wp(\omega) = \frac{1}{\omega^{2}} + a_{1}w^{2} + a_{2}w^{4} + \dots$$
Further substitution into the differential equation
$$\wp^{\prime\prime}(\omega) = 6(\wp(\omega))^{2} - \frac{g_{2}}{2}$$ yields
$$\frac{6}{\omega^{4}} +2b_{1} + \sum_{k=2}^{\infty}(2k)(2k-1)b_{k}\omega^{2k} = 6\left[ \frac{1}{\omega^{2}} + b_{1}\omega^{2} + \sum_{k=2}^{\infty}b_{k}\omega^{2k}\right]^{2} - \frac{g_{2}}{2}.$$
and the article here from mathworld.wolfram mentions the existence of a recurrence relation closed form solution given by equation (7). I figured I may be able to find all the coefficients in one swoop from achille's hint, but I cannot sniff out the next step to reproduce the relation.