a) Show that for any constant , the function (, ) = ^kx cos is a solution of Laplace's equation _xx + _yy= 0.

b) Show that for any constant , the function (, ) = ^{kxk^2y} is a solution of the heat equation _{xx y}= 0.

c) Show that for any constant , the function (, ) = ^kx-ky is a solution of the wave equation xx _yy = 0.

d) Show that for any constant , the function (, ) = (k/2) ² + (1 )( y²/2) is a solution of Poisson's equation _xx + _yy = 1.