The shortest universal solutions for non-linear dynamic equation

To study on the non-linear dynamic equation dx(t)/dt = F(x(t), t), the concept of “time-state space” — (t, x(t)) is introduced to obtain the shortest universal analytic solutions at any order series. F(x(t), t) can be expanded as Taylor series on independent variable t at the point of (t = 0, x(0)) in the time-status space. And then, the shortest universal analytic solutions at any order series can be obtained by integrating, and the convergence can also be proven.