A singular nonlinear differential equation arising in the Homann flow (Q1365097)

Existence and uniqueness of a positive solution of the singular nonlinear boundary value problem \[ yy''-\frac12(1-x^2)y'=0,\quad 0<x<1, \qquad y'(0)=-\frac12, \quad y(1)=0 \] is proved. First it is shown that the problem with \(y(1)=\frac1m\) instead of \(y(1)=0\) has a unique positive solution for each positive \(m\), and then it is shown that these solutions converge to a solution of the original problem if \(m\to\infty\).