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Show that the function satisfies the two-dimensional Laplace's equation $u_{x x}+u_{y y}=0 .$ $$ u=e^{-x} \cos y+e^{-y} \cos x $$...

Question

Show that the function satisfies the two-dimensional Laplace's equation $u_{x x}+u_{y y}=0 .$ $$ u=e^{-x} \cos y+e^{-y} \cos x $$

Show that the function satisfies the two-dimensional Laplace's equation $u_{x x}+u_{y y}=0 .$ $$ u=e^{-x} \cos y+e^{-y} \cos x $$



Answers

Verify that the function $ u = 1/\sqrt{x^2 + y^2 + z^2} $ is a solution of the three-dimensional Laplace equation $ u_{xx} + u_{yy} + u_{zz} = 0 $.


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