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!!!
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.
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.
