Question

u cos y sinh x, sin y cosh x satisfy laplace equation?

Answer 1

$$u=\cos y\sinh x+\sin y\cosh x$$

$$u_x=\cos y\cosh x+\sin y\sinh x$$

$$u_{xx}=\cos y\sinh x+\sin y\cosh x$$

$$u_y=-\sin y\sinh x+\cos y\cosh x$$

$$u_{yy}=-\cos y\sinh x-\sin y\cosh x$$

Laplace equation

$$u_{xx}+u_{yy}=0$$

Check

$$\cos y\sinh x+\sin y\cosh x$$

$$+(-\cos y\sinh x-\sin y\cosh x)=0,True$$

Therefore the function $u=\cos y\sinh x+\sin y\cosh x$ satisfies the Laplace equation.