# Difficult non-linear system of two differential equations

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• Mar 26th 2011, 08:22 AM
derdack
Difficult non-linear system of two differential equations
Hello people! ;) Here is the problem. Some ideas for analytical solutions of system of equations? Ai constants.

A1*x1''*x2*Cos(x1) - A2*x''*Sin(x1)- A3*x1'' + A4*((x2')^2)*Cos(x1) + A5*Sin(x1)=0

-A6*x1''+A7*x1''*x2*Cos(x1)-A8*x1''*Sin(x1)-A9*((x1')^2)x2*Sin(x1)-A10*((x1')^2)*Cos(x1)+A11=0

We know x1(0), x2(0), x1'(0), x2'(0).

Thank you in advance!

Kind regards!
• Mar 26th 2011, 11:09 AM
Jester
Are all the constants $a_i, i = 1 \, \cdots \, 11$ nonzero and truely arbitrary?
• Mar 26th 2011, 11:34 AM
derdack
Yes
• Mar 26th 2011, 11:42 AM
Jester
Are you interested in analytical solutions in which some of the constants are zero or related to one another?
• Mar 26th 2011, 11:48 AM
derdack
Constants are constants like in all of differential equations. All of them are non-zero. I need some of ideas for solving using some analytical method.
• Mar 26th 2011, 11:57 AM
derdack
In first equation it should be A2*x2''*Sin(x1). Constants are linearly independent.
• Mar 26th 2011, 12:02 PM
derdack
Dear Danny,

This is a fullsimplify form of equations where I replace the constants with Ai. All of them must be in equations. Thank you a lot for take this into consideration. :)
• Mar 26th 2011, 12:11 PM
derdack
Maybe I can find numerical solutions, but only for determined constants. In this case I can't see how constants affect on solutions. Because of that I need analytical method for solving. But how???
• Mar 27th 2011, 03:00 PM
derdack
What do you think about perturbation method after approximation of Sin and Cos with polynomial form?
• Mar 29th 2011, 07:36 AM
derdack
I have forgotten to say that constants Ai are known. I will find how x1[t] and x2[t] depend of [t, A1,A2,A3...] like solution. Is it so hard?