Section 14.3 Partial Derivatives

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. \(u = e^{-x} \cos y - e^{-y} \cos x\)