Solving a problem proposed by T. Ogawa (Q1941140)

The author studies the behaviour of the unique solution to the Cauchy problem \[ \begin{gathered} -\psi''(r)-{2\over r}\psi'(r)=c(\psi^3(r)+1),\quad r>0, \\ \psi(0)=\alpha,\quad \psi'(0)=0,\end{gathered} \] where \(c>0\) and \(\alpha\) is an arbitrary non-negative real number. The first zero of the radially symmetric solution is studied as a function of its value at the origin.