The solution of Jiachang coefficient linear differential equation by eigenvalue method sum up the linear differential equation with constant coefficients

1. Introduction 2. Preparation Knowledge 3. The solution 3.2 Euler equation of constant coefficient homogeneous linear differential equation and Euler equation 3.1 constant coefficient homogeneous linear differential equation 4. Non-homogeneous linear differential equation (comparison coefficient method) 4.1 form I 4.2 form II 4.3 Euler equation Another solution reference

1. Introduction

In this paper, the eigenvalue method of linear differential equation with constant coefficients is summarized. In section 4.2 of the literature [1], the solution of constant coefficient linear differential equation is introduced in detail. The various cases of the root of the eigenvalue equation (real roots or the number of roots & roots) are classified and explained, but because the classification is too careful, which makes the readers difficult to remember the root situation, this paper aims to unify the various cases of the characteristic root. It is convenient to remember the solution of differential equation. 2. Preparation of Knowledge

All the research in this section revolves around the equation
DNXDTN+A1 (t) dn−1xdtn−1+⋯+an−1 (t) dxdt+an (t) x=f (x) (1) d n x D t n + a 1 (t) d n−1 x D t n−1 +⋯+ a n−1 (t) d x D T + a N (t) x = f (x) (1) \frac{d^nx}{d t^n}+a_1 (t) \frac{d ^{n-1}x}{d t^{n-1}}+ \cdots +a_{n-1} (t) \frac{d x}{d T}+a_ N (t) x=f (x) \qquad (1)
Carried on. where AI (t) (I=1,2,⋯,n) a I (t) (i = 1, 2, ⋯, n) a_i (t) (I=1,2,\cdots,n) and F (t) f (t) are all interval [a,b] [A, b] [A, b] The continuous function on.
If {} f (t) ≡0 f (t) ≡0 f (t) \equiv 0, the equation (1) becomes