c2z2=a2x2+b2y2,
2c2z∂z=2a2x∂x=2b2y∂y, that is, c2z∂z=a2x∂x=b2y∂y
(1) c2z∂z/∂x=a2x, c2z∂z/∂y=b2y,
Rearranging (1): c2∂z/∂x=a2x/z, c2∂z/∂y=b2y/z,
c2z∂2z/∂x2+c2∂z/∂x=a2, c2z∂2z/∂y2+c2∂z/∂y=b2, (differentiating equations (1) again)
c2z∂2z/∂x2+a2x/z=a2, c2z∂2z/∂y2+b2y/z=b2, (substituting for c2∂z/∂x and c2∂z/∂y)
∂2z/∂x2+a2x/c2z2=a2/c2z, ∂2z/∂y2+b2y/c2z2=b2/c2z,
∂2z/∂x2+∂2z/∂y2+(a2x+b2y)/(a2x2+b2y2)=(a2+b2)/c2z, (substituting a2x2+b2y2 for c2z2).
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