How Are Derivatives Interrelated in ODEs and PDEs?

[Jhenrique]

Hellow everybody!

A form how various ODE are intercorrelated can be sinterized like this:

$t = t$
$y = y(t)$
$y' = y'(t,\;y)$
$y'' = y''(t,\;y,\;y')$
$y''' = y'''(t,\;y,\;y',\;y'')$

Until here, no problems!

But, how is such relationship wrt the PDE?

Would be this:

$x = x$
$y = y$
$u = u(x,\;y)$
$u_x = u_x(x,\;y,\;u)$
$u_y = u_y(x,\;y,\;u)$
$u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y)$
$u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y)$
$u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y)$
$u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y)$

Ie, the derivative of order n wrt x no depends of the derivative of order n wrt y (and vice versa), or depends? If depends, so the relationship would be this:

$x = x$
$y = y$
$u = u(x,\;y)$
$u_x = u_x(x,\;y,\;u,\;u_y)$
$u_y = u_y(x,\;y,\;u,\;u_x)$
$u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{yy},\;u_{xy},\;u_{yx})$
$u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{xy},\;u_{yx})$
$u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{yx})$
$u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{xy})$

Wich 2 last relation is correct?

 

[Jhenrique]

BTW, my question make sense?