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**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 )....

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