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## Homework Statement

The problem is to solve

[tex]\phi_{yy}-c^2 \phi_{xx} = 0[/tex]

[tex]\phi_y (x,0) = f'(x), x>0[/tex]

[tex]\phi_x (0,y) = \phi(0,y) = 0, y>0[/tex] or [tex]y<0[/tex]

## Homework Equations

The solution, before applying boundary conditions is obviously

[tex]\phi(x,y)=F(x+c y)+G(x-cy)[/tex]

## The Attempt at a Solution

I start with the general solution

[tex]\phi(x,y)=F(x+c y)+G(x-cy)[/tex]

and apply the two vanishing boundary conditions

[tex]\phi(0,y)=F(c y)+G(-cy)=0[/tex] or

[tex]1) F(\omega)+G(-\omega)=0[/tex]

[tex]\phi_x(0,y)=F'(c y)+G'(-cy)=0[/tex] or

[tex]2) F'(\omega)+G'(-\omega)=0[/tex]

Take the derivative of equation 1:

[tex]F'(\omega)-G'(-\omega)=0[/tex]

So [tex]F'[/tex] and [tex]G'[/tex] both vanish. Then how do we apply the first boundary condition?