The authors investigate the application of the known variational method to find the approximate solution of the strongly nonlinear differential equation of the form \[ \ddot{u}(t)+ \alpha u(t) + N(t,u(t), \dot{u}(t),\ddot{u}(t))=0, \qquad t\in (0,T], \] with the initial conditions \(u(0)=A\), \(\dot{u}(0)=B \), where \(\alpha >0\), \(N(t,u(t), \dot{u}(t),\ddot{u}(t))\) is a nonlinear function.
The variational iteration method \[ u_{n+1}(t)= u_{n}(t)+ \int^{t}_{0}\lambda (t,x) [\ddot{u}(x)+ \alpha u(x) + N(x,u(x), \dot{u}(x),\ddot{u}(x))]dx \] is applied, using the system of Chebyshev polynomials, to approximate the \( u_{n}(t)\). The approximate solution on the interval \((0,T]\) is obtained decomposing \([0,T]\) into non-overlapping intervals \(\Omega_{r}=[T_{r-1},T_{r}]\), \(r=1,2,\dots,m\); \(\Delta_{r}= T_{r}-T_{r-1}\). Numerical results for some examples are compared with the results given by Runge-Kutta method of fourth order.