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Math Help - artial Differential Equations Help!

  1. #1
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    artial Differential Equations Help!

    Show that u=f(2x+y^2)+g(2x-y^2) satisfies the equation y^2 d^2u/dx^2 + (1/y) du/dy - d^2u/dy^2=0 where f and g are arbitrary (twice differentiable) functions.

    I came up with fxx=0 fyy=2 gxx=0 gyy= 2. I think I need to come up with ux, uy, uxx and uyy but I'm not sure I am coming up with the correct answers. Can someone just show me how to do uxx and I can verify if I am doing it correctly. Thanks!
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    Quote Originally Posted by walter9459 View Post
    Show that u=f(2x+y^2)+g(2x-y^2) satisfies the equation y^2 d^2u/dx^2 + (1/y) du/dy - d^2u/dy^2=0 where f and g are arbitrary (twice differentiable) functions.

    I came up with fxx=0 fyy=2 gxx=0 gyy= 2. I think I need to come up with ux, uy, uxx and uyy but I'm not sure I am coming up with the correct answers. Can someone just show me how to do uxx and I can verify if I am doing it correctly. Thanks!
    Rmember that both f and g are arbitrary so

    u_x = 2 f'(2x+y^2) + 2 g'(2x-y^2),\;\;\;u_y = 2y f'(2x+y^2) - 2y g'(2x-y^2)

    Simlarly for
    u_{xx},\;u_{xy},\;\;u_{yy}
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