How to Solve the Laplace Equation for Potential Flow Around a Sphere?

[happyparticle]

Homework Statement

Consider the steady flow pattern produced when an impenetrable rigid spherical obstacle is placed in a uniformly flowing, incompressible, inviscid fluid. Find a solution for the potential flow around the sphere.

Relevant Equations

$\nabla^2 \phi = 0$

I tried to find a solution to the Laplace equation using spherical coordinates and the separable variable method. However, I found equations that I simply don't know how to find a solution. Thus, I tried in cylindrical coordinates with an invariance in $\theta$ but now I'm facing this equation.

$\frac{1}{s} \frac{d}{ds}(s \frac{dS}{ds}) = -k^2 S$

Is there a fairly simple solution for it or should I find another way to do this problem?

 

[anuttarasammyak]

I am not sure to distinguish S and s in your equation. Let me rewrite it with x for s and y for S
$$\frac{1}{x} \frac{d}{dx}(x \frac{dy}{dx}) = -k^2 y$$
Is it the right equation which has constant k with its physical dimension of [1/s] ？ It belongs to Sturm -Liouville equation whose solutions are Bessel functions.