In order to compute the holographic entanglement entropy for a spherical subsystem in AdS using the Ryu-Takayanagi conjecture, one needs to solve the following second order nonlinear differential equation:
$$zr(z)r''(z) - (d-1)r(z)(r'(z))^3 - (d-2)z(r'(z))^2 - (d-1)r(z)r'(z) - (d-2)z = 0$$
and obtain the solution $r^2 + z^2 = l^2$, I need help in obtaining this solution.
N.B.: The equation (in a mildly different form), and its solution are given in hep-th/0605073