# Thread: artial Differential Equations Help!

1. ## 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!

2. Originally Posted by walter9459
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
$u_{xx},\;u_{xy},\;\;u_{yy}$