Question

For the Laplace equation of radial symmetry, the Laplacian in polar coordinate

∇²u= ∂²u/∂r² + r∂u/r1∂ =0 In other words, you can assume u=u(r) and solve this ODE. Solve this to find solution of u.

Answer 1

The Laplace equation with radial symmetry is solved by integrating twice, yielding a general solution u(r) = C1 • ln(r) + C2, where C1 and C2 are constants determined by boundary conditions.

Step-by-step explanation:

The given differential equation for a function with radial symmetry is ∇2u = ∂²u/∂r² + (1/r)∂u/∂r = 0, where u = u(r). This is a second-order ordinary differential equation (ODE) that can be solved to find the radial function u(r). To solve this ODE, we will separate and integrate twice.

Steps to Solve the Laplace Equation

1. Integrate the first term ∂²u/∂r² which yields A + B/r, where A and B are integration constants.
2. Using the second term, we set A + B/r equal to zero and solve for u(r).
3. The general solution to the ODE is u(r) = C1 • ln(r) + C2, where C1 and C2 are arbitrary constants determined by boundary conditions.

Note that in the context of electrostatics, physics, or other applications, additional conditions or interpretations may be required to find a specific solution.