# Differential Equation satisfied by Weierstrass's Elliptic Function

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## Theorem

- $\paren {\dfrac {\d f} {\d z} }^2 = 4 f^3 - g_2 f - g_3$

where:

- $\ds g_2 = 60 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^4}$

and:

- $\ds g_3 = 140 \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^6}$

has the general solution:

- $\map f z = \map \wp {z + C; \omega_1, \omega_2}$

where:

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

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## Proof

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## Sources

- 1920: E.T. Whittaker and G.N. Watson:
*A Course of Modern Analysis*(3rd ed.): $20.22$: The differential equation satisfied by $\map \wp z$