Solving the PDE $au_x+bu_y+cu=0$ The PDE is transformed by the coordinate method via,
$\begin{cases}x'=ax+by\\y'=bx-ay\\\end{cases}$.
What I don't understand is how should I know I have to pick this as the transformation ?
How do I know that this is the transformation to be used?
Is it something to do with linear transformations in matrix?
You should recognize $au_x+bu_y$ as the dot product of $\nabla u$ with the vector $\left<a,b\right>$. Also known as the directional derivative. This makes the direction of vector $\left<a,b\right>$ seem special for this equation, so we take a coordinate system in which it is one of two coordinate axes. This is best done by a rotation matrix, or a multiple of it, so that the axes remain nicely orthogonal (although this isn't strictly necessary).