Problem with $y'=x^2+y^2$ when $y(0)=0$

Question

Referring to my question "Simple non-linear differential equation" and the associated answers and comments, the provided solutions in terms of fractional order Bessel functions or $\frac12$ order parabolic cylinder functions all seem to have a problem with requiring $y(0) = 0$. Never the less the series $$y = \frac{x^3}{3} + \frac{x^7}{63} + \frac{2x^{11}}{2079} + ..... $$ is a solution with $y(0) = 0$ so can anybody please explain why the Wolfram-solutions fail at $x =0$?

Answer 1

If in the solution given by Wolfram you let $C[1]\to\infty$, you obtain the solution wth $y(0)=0$: $$ y(x)=-\dfrac{x^2\,J_{-\frac{5}{4}}\Bigl(\dfrac{x^2}{2}\Bigr)-x^2\, J_{\frac{3}{4}}\Bigl(\dfrac{x^2}{2}\Bigr)+J_{-\frac{1}{4}}\Bigl(\dfrac{x^2}{2}\Bigr)}{2\,x\,J_{-\frac{1}{4}}\Bigl(\dfrac{x^2}{2}\Bigr)}. $$

Answer 2

Wolfram now shows the answer together with sample graphs (which it not did before). .

Answer 3

I have tried to use the Bessel functions recurrence relations with respect to order and obtained this simplyfied expression which does approach x^3/3 for small x. Further, I have veryfied by hand that it is actually the solution. PROBLEM SOLVED!