n階線性非齊次微分方程初值問(wèn)題的矩陣解法

曹玉平

(連云港職業(yè)技術(shù)學(xué)院基礎(chǔ)部,江蘇連云港 222006)

n階線性非齊次微分方程初值問(wèn)題的矩陣解法

曹玉平

(連云港職業(yè)技術(shù)學(xué)院基礎(chǔ)部,江蘇連云港 222006)

借助矩陣指數(shù)函數(shù)和狀態(tài)轉(zhuǎn)移矩陣的概念,結(jié)合線性代數(shù)和微分方程的有關(guān)結(jié)論,給出了 n階線性非齊次微分方程初值問(wèn)題的矩陣解法。

線性非齊次微分方程;初值問(wèn)題;矩陣解法

常微分方程有著深刻而生動(dòng)的實(shí)際背景,是現(xiàn)代科學(xué)技術(shù)中分析問(wèn)題與解決問(wèn)題的一個(gè)強(qiáng)有力的工具。本文借助矩陣和微分方程的有關(guān)結(jié)論,給出了n階線性非齊次微分方程初值問(wèn)題的矩陣解法。

0 預(yù)備知識(shí)

定義4設(shè) a1(t),a2(t),…,an(t),u(t)均為[a,b]上連續(xù)的已知函數(shù),且 u(t) ≠0,則稱 yn+ a1(t)y(n-1)+ a2(t)y(n-2)+…+an(t)y=u(t)為n階線性變系數(shù)非齊次微分方程。稱 yn+a1(t)y(n-1)+a2(t)y(n-2)+…+an(t)y=0為n階線性變系數(shù)齊次微分方程。[4]

1 n階線性非齊次微分方程初值問(wèn)題的矩陣解法

證明 將X中每個(gè)分量xi(t) (i=1,2,…,n)展開為Maclaurin級(jí)數(shù)得:

[1] 程云鵬.矩陣論[M].西安:西北工業(yè)大學(xué)出版社, 1999:149-150.

[2] 李新,何傳江.矩陣?yán)碚摷捌鋺?yīng)用[M].重慶:重慶大學(xué)出版社,2005:96.

[3] 劉丁酉.矩陣分析[M].武漢:武漢大學(xué)出版社,2003:199.

[4] 王高雄,周之銘,朱思銘,等.常微分方程[M].2版.北京:高等教育出版社,1983:102-105.

[5] Gene H Golub,charles FVan Loan.Matrix Computations [M].Baltimore:The Johns Hopkins University Press, 1983:107-124.

[6] 宣立新.高等數(shù)學(xué):上冊(cè)[M].北京:高等教育出版社. 2002:191192.

(責(zé)任編校:夏玉玲)

A Matrix Solution to Differential Equation Initial Value Question of n Steps Wrong Uniform Number Linear Constant Index

CAO Yu-ping
(Department of Basic Courses,Lianyungang Vocational and Technical College,Lianyungang 222006,China)

A matrix solution has been worked out to n-step linear nonhomogeneous differential equation initial value by means of matrix index function and the concept of state shift matrix as w ell as the relevant conclusions regarding linear algebra and differential equation.

linear nonhomogeneous differential equation;initial-value question;matrix solution

O175

A

1672-349X(2010)06-0017-03

2010-07-01

曹玉平(1963-)男,副教授,主要從事工程數(shù)學(xué)的教學(xué)和研究。