Hi

Refer this RG thread-

https://www.google.co.in/amp/s/www.researchgate.net/post/How_to_solve_a_set_of_coupled_nonlinear_ordinary_differential_equations_with_boundary_conditions_too_coupled/amp

First convert these three equations into six state equations using six state variables x1, x2, . . . x6. Then use e.g. MATLAB solver ode45. ode23s etc to solve it numerically. For your equations 0de23s seems to be more suitable.

Hope this helps

Thank you for the answer. Actually my equations has two second order derivatives. At least ODE45 needs than on the right side no derivative term should be there. So, I think it can not be applied directly.

As per one of my friend's suggestion, via algebraic manipulation, I will remove the 2nd order coupling and apply the suggested methods.

Thanks again

If some direct method is available, than please suggest.

The suggestion by your friend is correct. I have solved such types of coupled nonlinear equations of vibration problems with different combinations of state variables and getting the same results. So don't bother about the algebraic manipulations (but within the rules of mathematics! ). I think, this is the only simplest way.

Also you must take care while selecting the sampling frequency (time step) and length of simulation time. For this, it needs to linearize the equations near equilibrium and then find eigen frequencies and time constants.

Ok, thank you sir

I will have to see what is linearization for finding eigen frequency and time constant. Do you know about some literature regarding that. I will surely appreciate.

Refer to the book: Fundamentals of Vibrations by Meirovitch Leonard for linearization and finding eigen frequency. Then give a thought process that how can you make use of it to find sampling frequency (time step) and length of simulation time.

Ok I will read that content, thank you very much.

There are several good software that can solve these equation easily. I personally use Maple. dsolve in Maple does great job. Regardless of you want to solve numerically or analytically. Specifically if you want to try an exact solution use Maple by Lie group method, it is very powerful method.

I have used Mathematica, but not Maple. It is not available to me. I will have to see what I can do. I could not do it with Dsolve in Mathematica.

Thanks

@Nikul Jani,

Pleas consider that if you are searching for an analytical solution you maybe can not get any but if you want an numerical solution, I am quite sure that you can do it by Maple. I solved several nonlinear system of nonlinear ODEs with extremely nonlinear terms as well the system was stiff. If you have problem with Maple, let me know.

Ok Sir. Thank you

I understand that going for analytical solution can waste lot of time. I am also not searching for analytical solution. I will see how I can get Maple and learn.