Ex: 76

Determine whether each of the following functions is a solution of Laplace’s equation
\[u_{xx} + u_{yy} = 0.\]

  1. \(u = x^2 + y^2\)
  2. \(u = \frac{x^2 - y^2}{x^2 + y^2}\)
  3. \(u = x^3 + 3xy^2\)
  4. \(u = \ln \sqrt{x^2 + y^2}\)
  5. \(u = \sin x \cosh y + \cos x \sinh y\)
  6. \(u = e^{-x} \cos y - e^{-y} \cos x\)