1. ## Required Transformation...Canonical Form

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

The textbook says

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

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

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

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

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

2. Yes on both questions (but really only $u_{\eta \eta} = 0$ - divide by 4).