Simultaneous differential equation of second order

[player1_1_1]

Hello
sorry for my English, i know its bad;)
I have a simultaneous differential equation of second order (moving of point mass around point mass M in beginning of cartesian system)
\begin{cases}\frac{\mbox{d}^2x}{\mbox{d}t^2}=-GMx\left(x^2+y^2\right)^{-\frac{3}{2}}\\ \frac{\mbox{d}^2y}{\mbox{d}t^2}=-GMy\left(x^2+y^2\right)^{-\frac{3}{2}}\end{cases}
and I need to find how x and y changes depending on t.
solving by substitution takes very long time, so other method which allows to solve it in shorter and more simply way would be appreciated;)
i am not sure if this thread is good for this topic, please move it if its bad location
thanks for your help;)

 

[Meir Achuz]

Newton solved those equations in polar coordinates. I have never seen them solved in Cartesian coordinates.

 

[player1_1_1]

yeah, i came to it that it must be in polar system, i solved it and everything is good:)