# Thread: Difficult non-linear system of two differential equations

1. ## 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).

Kind regards!

2. Are all the constants $\displaystyle a_i, i = 1 \, \cdots \, 11$ nonzero and truely arbitrary?

3. Yes

4. Are you interested in analytical solutions in which some of the constants are zero or related to one another?

5. 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.

6. In first equation it should be A2*x2''*Sin(x1). Constants are linearly independent.

7. 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.

8. 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???

9. What do you think about perturbation method after approximation of Sin and Cos with polynomial form?

10. 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?