Differential Equation with a Singular point/singularity 
This is one problem from my problem sheet that I have no idea how to do because our lecturer (in my opinion) did a terrible job of explaining. I would really appreciate if you could help me understand what is going on here.
In lecture we defined the following differential equation:
$$y'=\frac{G(x,y)}{F(x,y)}$$
We assumed that $F,G$ vanish at point $(x_0,y_0)$. So we approximated the behavior of the diff. equation at point $(x_0,y_0)$ as:
$$y'=\frac{cx+dy}{ax+by}$$
(I have no idea why we can just do that). Also, we defined a matrix:
$$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$
Then we applied a invertible linear transformation to it:
$$\begin{pmatrix} \zeta \\ \eta\end{pmatrix}=D\begin{pmatrix}x \\y\end{pmatrix}=\begin{pmatrix}d_{11} & d_{12} \\d_{21} &d_{22}\end{pmatrix}\begin{pmatrix} x \\ y\end{pmatrix} \iff \begin{pmatrix} x \\ y\end{pmatrix}=D^{-1}\begin{pmatrix} \zeta \\ \eta\end{pmatrix}$$
Our lecturer then said something about the eigenvalues of $A$ giving us information about the solutions.
How do I use this information to solve the problem I mentioned above. It seems to me that I am missing some crucial information
Can someone explain to me what these matrices are for? I don't see the goal/pupose of this whole exercise. Is this some standard mathematical procedure when it comes to dealing with singularities?
Edit: In case this question is incomprehensible or is missing some information/context please leave a comment.
Edit: I am still interested in any hints/suggestions you might have. Thanks.
 A: The following remarks try to explain the philosophy behind this problem.
Seeing your differential equation I set up a "concomitant" first order ODE  for a vector-valued function $t\mapsto \bigl(x(t), \>y(t)\bigr)$ of an auxiliary "time" variable $t$, as follows:
$$\eqalign{\dot x&=x-Py\cr \dot y&=Px+Qy\cr}\tag{1}$$
The solutions $t\mapsto \bigl(x(t), \>y(t)\bigr)$ of this system can be considered as parametric representations of certain curves $\gamma$ in the $(x,y)$-plane. If such a curve is written as  graph of a function $f_\gamma: \>x\mapsto y(x)$ then
$$y'(x)={dy\over dx}={\dot y\over\dot x}={Px+Qy\over x-Py}\ ,$$
which shows that $f_\gamma$ satisfies the differential equation given at the outset.
The above serves as a motivation to investigate the linear system $(1)$ whose matrix is given by
$$A=\left[\matrix{1 &-P \cr P &Q\cr}\right]\ .$$
Systems of the kind $(1)$ are a prime paradigm in the theory of ODEs. Let it suffice for the moment that diagonalization of the matrix $A$ allows to "separate the unknown variables": In the coordinates $(\xi,\eta)$ corresponding to the "eigenbasis" of $A$ the system $(1)$ simply reads
$$\dot\xi=\lambda_1\>\xi,\qquad \dot\eta=\lambda_2\>\eta$$
and has the solutions
$$\xi(t)=C_1 e^{\lambda_1 t},\qquad \eta(t)=C_2e^{\lambda_2 t}\ .$$
In the case of your problem a non-negligible amount of calculation will result, which I hesitate to go into at the moment.
