= Solution
Set
$$
F(z)=\wp'(z)^2-4\wp(z)^3+g_2\wp(z)+g_3.
$$
This is an <elliptic function>. Write $a=g_2/20$ and $b=g_3/28$ in the expansion from part (i). At a lattice point, the coefficient of $z^{-2}$ in $F$ is $g_2-20a=0$, and its constant coefficient is $g_3-28b=0$; there are no other singular terms. Every apparent pole is therefore a <removable singularity>, and $F$ extends to an entire elliptic function with value zero at the lattice points.
An entire elliptic function is bounded on a <fundamental parallelogram> and hence everywhere. By <Liouville theorem> it is constant, and its value at zero is zero. Thus $F\equiv0$, proving the <Weierstrass elliptic differential equation>
$$
\boxed{\wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3.}
$$
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