The method of defining characteristic roots is a general method to solve homogeneous linear differential equations with constant coefficients. The characteristic root method can also be used to find the general term formula of recursive sequence, and its essence is the same as that of differential equation. R*r+p*r+q is called the characteristic equation of a recursive sequence: a(n+2)=pa(n+ 1)+qan. In the method of differential equation, let the characteristic equation r*r+p*r+q=0 be r 1, r2. 1 If the real root r 1 is not equal to R2y = c1* e (r1x)+C2 * e (r2x) .2 If the real root r1= R2y = (c1) An=0, and the corresponding characteristic equation is: x 2+sx+t = 0. Let two be α ≠ β, an = k * α (n-. An = (kn+m) * α (n-2), where the values of k and m are solved, the equation (1) can be solved by substituting the values of A 1 and A2 into the above general formula. The sequence satisfies: An+2-4 * An+ 1+. To find the solution of the general term An: the characteristic equation is (x-2) 2 = 0, so α = β = 2 and let an = (kn+m) * α (n-2), so (k+m)/2 = 1, (2k+m)=2, and the solution is: m. Fibonacci sequence satisfies: an+2-an+ 1-an = 0, A 1 = So α = (1-√ 5)/2, β = (1+√ 5)/2 Let an = k *. Then k+m.k * (1-√ 5)/2+m * (1) R2 is an = c 1 * r 1 n+C2 * R2 n, where the constant c 1, C2 is determined by the initial value a. (2) c 1r 1 2+c2r2 2 = b2 If the characteristic equation has two equal real roots r1= R2 = ran = (c 1+nc2) r n, where the constant c1,c2 is uniquely determined by the initial value. A simple solution to the equation that (1) A = (c1+C2) R (2) B = (c1+2c2) R2 has multiple characteristic roots. For constant coefficient homogeneous linear differential equation dX/dt=AX, when the primary factor of mi corresponding to characteristic root λ I (I = 65438) is (λ-λ I) Ki 1, …), (λ-λ I) Kimi, Ki 1+…+Kimi = ni, it corresponds to. At this time, the degree of polynomial P(i)j(t) is less than or equal to Mi- 1, (mi = max {ki 1 …, Kimi}). Because mi is difficult to calculate, this paper uses the characteristics of similarity matrix and Jordan standard form to find a convenient one between mi- 1 and ni- 1.