Need help with math homework, about Laplace's equation (fxx + fyy = 0)?

Substitute them in and see what happens.

(1) u = x^2 - y^2

u_x = 2x, u_xx = 2

u_y = -2y, u_yy = -2

==> u_xx + u_yy = 0.

This is a solution of Laplace's Equation.

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2) u = ln (sqrt(x^2 + y^2)) = (1/2) ln(x^2 + y^2).

u_x = x/(x^2 + y^2), u_xx = (y^2 - x^2) / (x^2 + y^2)^2

u_y = y/(x^2 + y^2), u_yy = (x^2 - y^2) / (x^2 + y^2)^2

==> u_xx + u_yy = 0.

This is a solution of Laplace's Equation.

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3) u = e^(-x) cos y - e^(-y) cos x

u_x = -e^(-x) cos y + e^(-y) sin x, u_xx = e^(-x) cos y + e^(-y) cos x

u_y = -e^(-x) sin y + e^(-y) cos x, u_yy = -e^(-x) cos y - e^(-y) cos x

==> u_xx + u_yy = 0.

This is a solution of Laplace's Equation.

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4) u = x^2 + y^2

u_xx + u_yy = 4; this is not a solution.

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5) u = x^3 + 3xy^2

u_xx + u_yy = 12x; this is not a solution.

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6) u = sin x cosh y + cos x sinh y

u_x = cos x cosh y - sin x sinh y, u_xx = -sin x cosh y - cos x sinh y

u_y = sin x sinh y + cos x cosh y, u_yy = sin x cosh y + cos x sinh y

==> u_xx + u_yy = 0.

This is a solution of Laplace's Equation.

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I hope this helps!

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