# New system of two second order differential equations

I know one more thing from physical system. If we can assume the solutions in form $$x=Ae^{jp_1t}, \quad y=Be^{jp_2t}, \quad j=-1^{1/2}$$ I know that $$p_1=2p_2$$

If someone can help me. It is need to find a analytic or numeric solutions where $D_i$ are known constants. If the system can describe by lower number of constants and lower order, how can I get a numerical solutions in function of this constants using some of methods (perturbation or some software - Mathematica)

$$D_1x''+D_2y''(x'-y')-D_2x'y'+D_3x=0$$

$$D_4y''+D_2x''(x'-y')+D_2x'y'+D_5=0$$

Initial conditions

$$x(0)=a, \quad y(0)=0, \quad x'(0)=0, \quad y'(0)=0$$

where $(')=d/dt, ('')=d^2/dt^2$.

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@reptil: please note that your mathematics display does not automagically becomes TeX-ified. Please take the time to learn "How to write math" (look to your right! it is in a box!), so other users won't have to do it for you (again) in the future. –  Willie Wong Apr 12 '11 at 17:04
You can get numerical solutions for a given set of constants by time-stepping using an appropriate numerical scheme (fourth-order Runge-Kutte is popular, and implemented in most numerical mathematics packages, eg Matlab, Scilab, Octave). You can then use a numerical continuation software package (eg AUTO) to explore the solution as the constants vary. –  Chris Taylor Apr 13 '11 at 11:08
What do you think about a way which Hans Engler describe in my first post. This can be analytical way (not complete, but helpful). If you can take this problem into consideration to reduce number of constants to get semi-analytical solutions, I will be very grateful. 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 dear Chris. –  reptil Apr 13 '11 at 14:12