= Solution
Write the analytic solution as $w(z)=\sum_{n\geq0}a_nz^n$. The constant term in the differential equation gives $a_1=0$, and matching the coefficient of $z^{n+1}$ gives
$$
(n+2)^2a_{n+2}+4a_n=0,\qquad n\geq0.
$$
All odd coefficients therefore vanish; $a_2=-a_0$ and $a_4=a_0/4$. The first three nonzero terms are
$$
\boxed{w(z)=a_0\left(1-z^2+\frac{z^4}{4}+O(z^6)\right).}
$$
In full, $a_{2j}=(-1)^ja_0/(j!)^2$, so $w=a_0J_0(2z)$, a <Bessel function of the first kind>. The <ratio test> proves convergence for every complex $z$. The other independent solution has a logarithmic singularity at zero and does not satisfy the analyticity requirement.
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