Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 8 / b / Solution

Write the analytic solution as $w(z)={\sum }_{n\ge 0}{a}_n{z}^n$. The constant term in the differential equation gives $a_1=0$, and matching the coefficient of $z^{n+1}$ givesAll odd coefficients therefore vanish; $a_2=-a_0$ and $a_4=a_0/4$. The first three nonzero terms areIn full, $a_{2j}=(-1{)}^j{a}_0/(j!{)}^2$, so $w=a_0{J}_0(2z)$, 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.