18.3.3.1 Solution ppe-a

Question:

Show that the given solution

$$u(x,y) = \sin(nx)\cosh(ny)$$

for natural $n$ does indeed satisfisfy the Laplace equation

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

and the boundary conditions

$$u(x,0)=\sin(x)\quad u_y(x,0)=0 \quad u(0,y)=0 \quad u(\pi ,y)=0$$

Answer:

Plug it in.