Any order approximate solution of the state equation for an affine nonlinear system

The state equation of an typical affine nonlinear system was solved with the ordinary differential equation theory. By utilizing the expansion expression of equilibrium point of the system, the homogeneous equation's solution was obtained, and then the nonlinear differential equation was equivalent to its nonlinear Volterra's integral equation of the second kind by the constant variation method. Any order approximate solution of the equation was presented, and its convergence was mathematically proved by the successive approximation method.