Question: Classify the equation $\displaystyle u_{xx} + 4u_{xy} + 4u_{yy} = 0$, reduce it to standard form, and state the required transformation.

The textbook says

$\displaystyle D= b^2 - 4ac = 16-4*4 = 0 \ \Rightarrow \ $ Parabolic

Then, $\displaystyle \xi = bx - 2ay = 4x - 2y$ and $\displaystyle \eta = y$

So the equation reduces to $\displaystyle 4u_{\eta \eta} = 0$

which all makes sense, but I am a little confused with what is the transformation and what is standard form.

Is [mATH]\xi = bx - 2ay = 4x - 2y[/tex] and $\displaystyle \eta = y$ the required transformation and is $\displaystyle 4u_{\eta \eta} = 0$ standard form?