  In the present paper, the nonlinear differential equation of pendulum is
investigated to find an exact closed form solution, satisfying governing
equation as well as initial conditions. The new concepts used in the suggested
method are introduced. Regarding the fact that the governing equation for any
arbitrary system represents its inherent properties, it is shown that the
nonlinear term causes that the system to have a variable identity. Hence, the
original function is included as a variable in the solution to can take into
account the variation of governing equation. To find the exact closed form
solution, the variation of the nonlinear differential equation tends to zero,
where in this case the system with a local linear differential equation has a
definite identity with a definite local answer. It is shown that the general
answer is an arbitrary curve on a surface, a newly developed concept known as
super function, and different initial conditions give different curves as
particular solutions. The comparison of the results with those of finite
difference shows an exact agreement for any arbitrary amplitude and initial
conditions.
