As you may know the most common method will be to use the implicit Euler method for solving an ODE. You may also look into the central difference method or the midpoint method.

You have a non-linear system of equations in two variables. You will need to write a function that returns the derivatives of theta and psi, in terms of theta, psi and the other parameters. If you have access to MATLAB, I'd recommend using ode15s. If not, the scipy package in python also provides a library of ode solvers that is very easy to use. Look for scipy.integrate.odeint.

Hello manmatha krishnan Palraj

In the first instance, your coupled system is nonlinear because of the terms cosψ(ψt) and cosψ(θt)

Secondly you did not specify the initial conditions.

The problem can be solved with Newmark's θ= 1/2, β = 1/4 method applied to Eqs 1 and 2 sequentially.

we can approximate cosψ =1up to 1 rad..i.e 58 deg..so the equation will became linear and how to solve this..?

Hello manmatha krishnan Palraj

See the attached pdf for clarification.

thanks Amaechi J. Anyaegbunam for your suggestion..i will try for newmark's methaod

You can use the splitting method.

@ALL

Use MATLAB for analytical or numerical method .

B.Rath