# How to solve the equation of motion of the minimal surface for spherical subsystems in AdS?

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

In the case of a disk $$d=2$$, the equation is simplified: $$zr(z)r''(z) - r(z)(r'(z))^3 - r(z)r'(z) = 0$$ We discard the obvious solution of the equation $$r = 0$$, then putting $$r' = y$$ we find $$zy'=y^3+y$$ This equation is easily integrated, and using the boundary condition $$y(R) =\infty$$, we find $$\frac {r'}{\sqrt {1+r'^2}}=\frac {z}{R}$$
Solving the equation for $$r$$, we have $$r'=\pm \frac {z}{\sqrt {R^2-z^2}}$$ And find finally $$r=\pm \sqrt {R^2-z^2}$$ Note that in the article Aspects of Holographic Entanglement Entropy the authors cited another equation on p.34 $$rzz′′ + (d−1)z(z′)^3 + (d−1)zz′ + dr(z′)^2 + dr = 0.$$ For this equation, the solution $$z^2+r^2=R^2$$ exists for any $$d$$.
• @SabyasachiMaulik Use code r*z*z'' + (d \[Minus] 1) z (z')^3 + (d \[Minus] 1) z*z' + d*r (z')^2 + d*r /. {z -> Sqrt[R^2 - r^2], z' -> -(r/Sqrt[-r^2 + R^2]), z'' -> -(r^2/(-r^2 + R^2)^(3/2)) - 1/Sqrt[-r^2 + R^2]} // Simplify` – Alex Trounev May 13 at 15:48