system of two second order differential equations Hi everyone, I know that this system dont have analytical solutions. I want to get numerical solutions, but in function of some constants $A_i$. Mathematica can help me, but if somebody have idea? This equations describe a physical model 
$$ A_6 x + A_4 (y')^2 - 2 A_2 x'' - A_3xy'' + A_4yy'' = 0$$
$$ A_5 - A_3 (x')^2 - A_3xx'' + A_4yx'' - 2A_1y'' = 0$$
$'=(d/dt)$, $''=(d^2/dt^2)$, $A_i$-known constants. The initial conditions are:
$$ x(0)=a, y(0)=0, x'(0)=0, y'(0)=0$$
Thank you in advance!!! 
 A: You can reduce the number of parameters quite a bit, for starters. Set 
$$x = \alpha u , \; y = \beta v, \; t = \gamma \tau
$$
and write $\dot w = \frac{d}{d \tau} w, \ddot w = \frac{d^2}{d\tau^2} w$. By choosing the constants $\alpha, \beta, \gamma$ properly, you should be able to nondimensionalize the system to something like
$$c u + (\dot v)^2 - \ddot u - d u \ddot v + v \ddot v = 0$$
$$1 - (\dot u)^2 - u \ddot u + d^{-1} \ddot u v - \ddot v = 0 $$
$$
u(0) = \tilde a, \; v(0) = \dot u(0) = \dot v(0) = 0.
$$
So there are only three independent constants in the system, not 7.
A: It is unlikely that there is an analytic solution but you may be able to make some progress by rewriting as a first-order system. For example, with the equations as in Hans Engler's answer, you can define $w=\dot{u}$ and $z=\dot{v}$, and get a system of equations
$\dot{u} = w$
$\dot{v} = z$
$\dot{w} = \frac{cu + z^2 + (v-du)(1-w^2)}{1-(du-v)(u-v/d)}$
$\dot{z} = \frac{(v/d-u)(cu+z^2) + 1 - w^2}{1-(du-v)(u-v/d)}$
with
$u(0)=\tilde{a}$, $v(0)=w(0)=z(0)=0$.
This looks more complicated than the original set of equations, but you have the advantage that it's first-order and autonomous, and hence amenable to the techniques applicable to first-order autonomous nonlinear dynamical systems, such as linearization about fixed points, analysis of periodic orbits, energy theorems etc.
