# Radius of convergence to be proved more precisely (differential equation)

There is a differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.

It is possible to get a solution which is a power series (see below). However, I am looking for an analytical proof that the radius of convergence of the power series is near $7/2$

Full series can be obtained by substituting the formal power series expansion into the equation and matching the terms at equal powers of t. Coefficients of the resulting power series $r(t)=\sum_{n=1}^\infty R_{2n} t^{2n}$

can be computed using the recurrent formulae $R_2=k/2$, $R_4=-k/32$

$R_{2n}=-\frac {R_{2n-2}} {8n} - \frac {1} {4kn} \sum_{i=1}^{n-2} (R_{2i}+4(i+1)(n-i)R_{2i+2})R_{2n-2i}$, $n=3,4…$

Please note that numerical calculation results in that the radius of convergence is near $7/2$, but I need to be more precise. So, I hope to find an analytical approach and a proof. Any help (ideas) are highly welcomed.

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We can assume that $k=1$. Maybe it's simpler to write $a_k:=R_{2k}$, then use the recurrent relation to find the maximal $r$ such that if $|x|<r$ then $\sup_k|a_kx^k|$ is finite. –  Davide Giraudo Jan 17 '13 at 11:18