System of differential equation with variable coefficent How to solve this system of differential equations   $x'(t)=\frac{a+s}{(1-t)d}x(t)-\frac{b}{(1-t)d}y(t)$  and $y'(t)=\frac{a}{(1-t)d}x(t)-\frac{(s+b+(1-t)c)}{(1-t)d}y(t)$ where a,b,c,d and s are constants.
 A: The system of two first order ODE can be split into two independent ODEs of second order of hypergeometric kind. It is shown below that the analytic solution $x(t)$ involves the confluent hypergeometric function of second kind and the associated Legendre polynomial.
In a first part, some change of parameters and variable allows to simplify the equations in order to bring them on a more standard form :

In the second part, the second order ODE (with the unknown $x$ only) can be solved thanks to standard method, but arduous and rather boring. Thanks to WolframAlpha, the solution is shown below :

The second order ODE with the unknown $y$ only could be solved on the same manner (replace $k_4$ by $k_5$). The constants $c_1$ and $c_2$ must be changed to other constants, say $c_3$ and $c_4$. But they are related to $c_1$ and $c_2$ (The whole solution $x(t),y(t)$ of the system must include only two arbitrary constants, not four). The relationships between $(c_1,c_2)$ and $(c_3,c_4)$ will be difficult to be derived. Instead of solving the second ODE, it is easier to compute $y(t)$ directly from the last equation written at the end of the above page.
