Laplace's equation does not hold true at the location of a dipole; instead, Poisson's equation should be used to account for the dipole's charge distribution. The charge distribution for an ideal dipole involves a delta function and its derivative, indicating that the potential satisfies Laplace's equation everywhere except at the dipole itself. When solving for the electric potential in a dielectric sphere containing a dipole, one should apply Laplace's equation outside and Poisson's equation inside, while ensuring boundary conditions are met. The discussion highlights the importance of accurately defining charge distributions to avoid algebraic errors in calculations. Ultimately, the potential satisfies Poisson's equation everywhere, with the specified source charge density.