# 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!!!

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What is the "physical model" this system describes; i.e., where does it come from? Knowing this might help answer your question. – drbobmeister Apr 10 '11 at 18:03

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.

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Thank you a lot professor Engler. In my new post - New system of two second order differential equations I got maybe simple system but I am not sure can I get a solutions like this way. If you can help me it will be great. In some way, if the number of constants can reduce, can I final get the solutions [x(...,t),y(,...t)]? Thank you again! – reptil Apr 13 '11 at 8:37
If you can take this problem into consideration to reduce number of constants to get semi-analytical solutions for my new post, I will be very grateful (New system of two second order differential equations). Of course, to lead system to may cause long with analytical aid. But I have 5 constants and Runge-Kutte can not help me. Just for special cause, where constants have numerical value. Thank you professor Engler. – reptil Apr 13 '11 at 14:17

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.

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Thank you for good suggestion. If you can describe some of method on simple example or take me to some literature. In my new post - New system of two second order differential equations I got maybe simple system but I am not sure can I get a solutions like this way. If you can help me it will be great. In some way, if the number of constants can reduce, can I final get the solutions [x(...,t),y(,...t)]? Thank you a lot! – reptil Apr 13 '11 at 8:36
What do you think to find dw/dz and eliminate t? Can you help me to describe my new system of two second order differential equations to find w and z like here? Yes iti is trivial but I didn't understand how Hans Engler got just 1 in front of almost all (du/dt,d^2v/dt^2...) – reptil Apr 14 '11 at 11:19