Differential Equation satisfied by Weierstrass's Elliptic Function

Theorem
The differential equation:


 * $\left({\dfrac {\rd f} {\rd z} }\right)^2 = 4 f^3 - g_2 f - g_3$

where:


 * $\displaystyle g_2 = 60 \sum_{\left({n, m}\right) \mathop \in \Z^2 \setminus \left({0, 0}\right)} \frac 1 {\left({2 m \omega_1 + 2 n \omega_2}\right)^4}$

and:


 * $\displaystyle g_3 = 140 \sum_{\left({n, m}\right) \mathop \in \Z^2 \setminus \left({0, 0}\right)} \frac 1 {\left({2 m \omega_1 + 2 n \omega_2}\right)^6}$

has the general solution:


 * $f \left({z}\right) = \wp \left({z + C; \omega_1, \omega_2}\right)$

where:


 * $\wp$ is Weierstrass's elliptic function
 * $C$ is an arbitrary constant
 * $\omega_1$, $\omega_2$ are constants independent of $z$.