定義
特徵根法是解常係數線性微分方程的一種通用方法。
特徵根法也可用於通過數列的遞推公式(即差分方程,必須為線性)求通項公式,其本質與微分方程相同。
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![特徵根法](/img/a/bbf/wZwpmL1QzM1EDM0czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL0czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
稱為二階齊次線性差分方程: 加權的特徵方程。
利用特徵根法解方程
對微分方程:
![特徵根法](/img/d/1d6/wZwpmLyEjM2AjM2gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLyMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設特徵方程 兩根為 r、 r。
① 若實根 r不等於 r
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.
② 若實根 r= r
③ 若有一對共軛復根 a± bi
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對差分方程:
1) 若特徵方程有兩個不等實根 r、 r
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則
其中常數 c、 c由初始值 a= a、 a= b 唯一確定。
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(1)
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(2)
2) 若特徵方程有兩個相等實根 r= r= r
![特徵根法](/img/5/c65/wZwpmL3EDM5kDM4EDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL0IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
其中常數 c、 c由初始值唯一確定。
![特徵根法](/img/e/e22/wZwpmLwcDN3YTO2UzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
(1)
![特徵根法](/img/1/7c7/wZwpmLyIjN4YzN3IDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL4czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(2)
![特徵根法](/img/5/563/wZwpmLwIzMwIDN1gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLwAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
3 )若特徵方程有一對共軛復根,則有
![特徵根法](/img/2/160/wZwpmL3YzM0MTN2UzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL1UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![特徵根法](/img/c/c59/wZwpmL3YjNwgDN3EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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一類重特徵根對方程解的簡便解法
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![特徵根法](/img/f/c97/wZwpmLzEDMzYDNxIjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![特徵根法](/img/3/f48/wZwpmLyUzNyMTO2UjM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL1IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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![特徵根法](/img/5/460/wZwpmLxATN1gTOwYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![特徵根法](/img/9/91b/wZwpmLygDNxkTO3MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![特徵根法](/img/5/041/wZwpmLxcTMxMzN3EDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLxQzLxQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
對於常係數齊次線性微分方程組 ,當矩陣 A的特徵根 的重數是 ,對應的 m個初等因子是 , 時,它對應方程中ni個線性無關解,其結構形如 ,此時多項式 的次數小於等於 , 。
![特徵根法](/img/9/91b/wZwpmLygDNxkTO3MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![特徵根法](/img/0/3ac/wZwpmL3czM4UzN4czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzLyAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![特徵根法](/img/5/460/wZwpmLxATN1gTOwYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
由於 M計算起來非常困難,本文利用相似矩陣的特點和Jordan標準型在 與 之間找到了一個便於套用的多項式 次數的上界,使計算起來更加方便和有效。