# 特徵方程式

${\displaystyle a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots +a_{1}y'+a_{0}y=0,}$

${\displaystyle a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots +a_{1}r+a_{0}=0}$

${\displaystyle y_{t+n}=b_{1}y_{t+n-1}+\cdots +b_{n}y_{t}}$

${\displaystyle r^{n}-b_{1}r^{n-1}-\cdots -b_{n}=0,}$

## 推導

${\displaystyle a_{n}y^{(n)}+a_{n-1}y^{(n-1)}+\cdots +a_{1}y^{\prime }+a_{0}y=0}$

${\displaystyle a_{n}r^{n}e^{rx}+a_{n-1}r^{n-1}e^{rx}+\cdots +a_{1}re^{rx}+a_{0}e^{rx}=0}$

${\displaystyle a_{n}r^{n}+a_{n-1}r^{n-1}+\cdots +a_{1}r+a_{0}=0}$

## 有關通解的公式

${\displaystyle y(x)=y_{\mathrm {D} }(x)+y_{\mathrm {R} _{1}}(x)+\cdots +y_{\mathrm {R} _{h}}(x)+y_{\mathrm {C} _{1}}(x)+\cdots +y_{\mathrm {C} _{k}}(x)}$

### 例子

${\displaystyle y^{(5)}+y^{(4)}-4y^{(3)}-16y''-20y'-12y=0}$

${\displaystyle r^{5}+r^{4}-4r^{3}-16r^{2}-20r-12=0}$

${\displaystyle (r-3)\left(r^{2}+2r+2\right)^{2}=0}$

${\displaystyle y(x)=c_{1}e^{3x}+e^{-x}(c_{2}\cos x+c_{3}\sin x)+xe^{-x}(c_{4}\cos x+c_{5}\sin x)}$

### 相異實根

${\displaystyle y_{\mathrm {D} }(x)=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+\cdots +c_{n}e^{r_{n}x}}$

### 重根實根

${\displaystyle \left({\frac {d}{dx}}-r_{1}\right)^{k}y=0}$ .

${\displaystyle \left({\frac {d}{dx}}-r_{1}\right)ue^{r_{1}x}={\frac {d}{dx}}\left(ue^{r_{1}x}\right)-r_{1}ue^{r_{1}x}={\frac {d}{dx}}(u)e^{r_{1}x}+r_{1}ue^{r_{1}x}-r_{1}ue^{r_{1}x}={\frac {d}{dx}}(u)e^{r_{1}x}}$

${\displaystyle \left({\frac {d}{dx}}-r_{1}\right)^{k}ue^{r_{1}x}={\frac {d^{k}}{dx^{k}}}(u)e^{r_{1}x}=0}$

${\displaystyle {\frac {d^{k}}{dx^{k}}}(u)=u^{(k)}=0}$

${\displaystyle y_{\mathrm {R} }(x)=e^{r_{1}x}\left(c_{1}+c_{2}x+\cdots +c_{k}x^{k-1}\right)}$

### 複數根

{\displaystyle {\begin{aligned}y(x)&=c_{1}e^{(a+bi)x}+c_{2}e^{(a-bi)x}\\&=c_{1}e^{ax}(\cos bx+i\sin bx)+c_{2}e^{ax}(\cos bx-i\sin bx)\\&=\left(c_{1}+c_{2}\right)e^{ax}\cos bx+i(c_{1}-c_{2})e^{ax}\sin bx\end{aligned}}}

${\displaystyle y_{\mathrm {C} }(x)=e^{ax}\left(c_{1}\cos bx+c_{2}\sin bx\right)}$

## 參考資料

1. Edwards, C. Henry; Penney, David E. Chapter 3. Differential Equations: Computing and Modeling. David Calvis. Upper Saddle River, New Jersey: Pearson Education. 2008: 156–170. ISBN 978-0-13-600438-7.
2. Smith, David Eugene. History of Modern Mathematics: Differential Equations. University of South Florida. [2019-05-05]. （原始内容存档于2011-07-20）.
3. ^ Baumol, William J. Economic Dynamics 3rd. 1970: 172.
4. ^ Chiang, Alpha. Fundamental Methods of Mathematical Economics 3rd. 1984: 578, 600.
5. Chu, Herman; Shah, Gaurav; Macall, Tom. Linear Homogeneous Ordinary Differential Equations with Constant Coefficients. eFunda. [1 March 2011]. （原始内容存档于2019-10-24）.
6. Cohen, Abraham. An Elementary Treatise on Differential Equations. D. C. Heath and Company. 1906.
7. ^ Dawkins, Paul. Differential Equation Terminology. Paul's Online Math Notes. [2 March 2011]. （原始内容存档于2021-04-14）.