1
$\begingroup$

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!!!

$\endgroup$
1
  • 1
    $\begingroup$ What is the "physical model" this system describes; i.e., where does it come from? Knowing this might help answer your question. $\endgroup$ Apr 10, 2011 at 18:03

2 Answers 2

2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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! $\endgroup$
    – reptil
    Apr 13, 2011 at 8:37
  • $\begingroup$ 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. $\endgroup$
    – reptil
    Apr 13, 2011 at 14:17
2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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! $\endgroup$
    – reptil
    Apr 13, 2011 at 8:36
  • $\begingroup$ 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...) $\endgroup$
    – reptil
    Apr 14, 2011 at 11:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.