2nd derivative over 1st derivative squared & Dirichlet Problem

[Verdict]

Homework Statement

First of all, for a different problem, I have the following written down:
$$\frac{\frac{{\partial{}}^2\varphi{}}{\partial{}u^2}}{{\left(\frac{\partial{}\varphi{}}{\partial{}u}\right)}^2}$$
Where phi is a function of u, which is a function of x and y. Also, u is harmonic.
Now, I want to rewrite this in some suggestive form, if there is one.

Now, onto my other question:

Homework Equations

Uhm, I can't really think of any at this point. Maybe the rotation matrix to rotate the strip?

The Attempt at a Solution

Concerning the rewriting: I already thought of
$$\frac{\partial{}}{\partial{}u}\left(\frac{-1}{\frac{\partial{}\varphi{}}{\partial{}u}}\right)$$ but I somehow want it to be a function of phi, and I don't think this qualifies as such.

Alright, and then the dirichlet problem. In my course, we haven't done any complicated PDE's yet whatsoever, so I checked with the teacher and he doesn't want a general formula, just a specific solution that works.
For question a, I found a solution:
(a+b)/2 - (a-b)/2 * X
seems to satisfy the conditions given.

Now for B, I somehow need to rotate and translate and scale the strip to the same one as the first. However, I don't understand how. If I understand correctly, I need to 'move' the plane bounded by y = x and y = x + 2 to the plane bounded by x = -1 and x = 1.
In order to use the rotation matrix, I have to make y = x and y = x + 2 into a vector first though. I don't really understand how, so it would be great if someone could give me a hint!

 

[Verdict]

Alright, so for b this is what I have thought of so far. I need to apply the rotation-45-degree matrix, rescale, and then shift. As I have to rescale anyway, I'll just take the matrix to be
1 -1
1 1

Instead of the square root two factors. However, I don't really know which points to use to find out exactly how it is done. I suppose (0,0) of the line y=x is still (0,0) after the rotation, and this just has to be shifted by 1 in the x direction. Does this mean that my transformation is just Sqrt(2)*(Rotation matrix)*original vector + a vector (1,0)?
So then x' = x - y + 1 and y' = x + y

But how do I use this to solve the differential equation?