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!!!
Best Answer
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.