Notes-LinearSystems - Systems of First Order Linear Differential Equations We will now turn our attention to solving systems of simultaneous

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Systems of First Order Linear Differential Equations We will now turn our attention to solving systems of simultaneous homogeneous first order linear differential equations. The solutions of such systems require much linear algebra (Math 220). But since it is not a prerequisite for this course, we have to limit ourselves to the simplest instances: those systems of two equations and two unknowns only. But first, we shall have a brief overview and learn some notations and terminology. A system of n linear first order differential equations in n unknowns (an n × n system of linear equations) has the general form: x 1 ′ = a 11 x 1 + a 12 x 2 + … + a 1 n x n + g 1 x 2 ′ = a 21 x 1 + a 22 x 2 + … + a 2 n x n + g 2 x 3 ′ = a 31 x 1 + a 32 x 2 + … + a 3 n x n + g 3 (*) : : : : : : x n ′ = a n 1 x 1 + a n 2 x 2 + … + a nn x n + g n Where the coefficients a ij ’s, and g i ’s are arbitrary functions of t . If every term g i is constant zero, then the system is said to be homogeneous. Otherwise, it is a nonhomogeneous system if even one of the g ’s is nonzero. The system (*) is most often given in a shorthand format as a matrix-vector equation, in the form: x ′ = Ax + g + = ′ ′ ′ ′ n n nn n n n n n n g g g g x x x x a a a a a a a a a a a a x x x x : : : : ... : : : : : : : : ... ... ... : : 3 2 1 3 2 1 2 1 3 32 31 2 22 21 1 12 11 3 2 1 x′ A x g Where the matrix of coefficients, A , is called the coefficient matrix of the system. The vectors x ′, x , and g are x ′ = ′ ′ ′ ′ n x x x x : 3 2 1 , x = n x x x x : 3 2 1 , g = n g g g g : 3 2 1 . For a homogeneous system, g is the zero vector. Hence it has the form x ′ = Ax . Fact : Every n-th order linear equation is equivalent to a system of n first order linear equations. Examples : (i) The mechanical vibration equation m u ″ + γ u ′ + k u = F ( t ) is equivalent to m t F x m x m k x x x ) ( 2 1 2 2 1 +-- = ′ = ′ γ Note that the system would be homogeneous (respectively, nonhomogeneous) if the original equation is homogeneous (respectively, nonhomogeneous). (ii) y ″′ ¡ 2 y ″ + 3 y ′ ¡ 4 y = 0 is equivalent to x 1 ′ = x 2 x 2 ′ = x 3 x 3 ′ = 4 x 1 ¡ 3 x 2 + 2 x 3 This process can be easily generalized. Given an n-th order linear equation a n y ( n ) + a n ¡1 y ( n ¡1) + a n ¡2 y ( n ¡2) + … + a 2 y ″ + a 1 y ′ + a y = g ( t )....
View Full Document

This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

Page1 / 33

Notes-LinearSystems - Systems of First Order Linear Differential Equations We will now turn our attention to solving systems of simultaneous

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online