Functions satisfying differential equation of the Weierstrass elliptic function $\wp$

Question

The Weierstrass elliptic function $\wp$ satisfies the following differential equation: $${\wp'}^2 = 4 \wp^3 - g_2 \wp - g_3$$ Which other functions do?

Are the solutions always elliptic functions?

Answer 1

Given theorem on existence and uniquenes of solutions to ordinary differential equations (Picard's theorem) there is only one local solution if the Lipschitz condition is satisfied, in fact it is when we take $x_1$,$x_2$ in the same parallelogram (determined with $(g_2,g_3)$) away from the lattice points. Since $\wp$ is doubly periodic it determines the solution everywhere (the lattice points are the second order poles).

Rewriting the equation one can reformulate the definition of the Weierstrass elliptic function, i.e. $\wp(x;g_2,g_3)$ yields the value $z$ for which $$x=\int_{\infty}^{z} \frac{d t}{\sqrt{4t^3-g_2 t-g_3}} $$

Reassuming there is only one solution if an initial condition is prescribed appropriately and it is just $\wp(x\pm c;g_2,g_3)$, where $c$ can be determined from the initial condition. If we prescribe e.g. this condition $f(\infty)=0$, then there are no solutions unless $\;g_2=0=g_3$.

In a degenerate case $g_2=g_3=0$ the only solution is $f(x)=\frac{1}{\left(x \pm c\right)^2}$ and this may be considered as non-elliptic function since it is not periodic, elliptic functions are doubly periodic meromorphic functions in the complex plane.