Chapter 6 Slides - Systems of Linear Dierential Equations I Converting a linear equation to a linear system Consider the third-order equation y or

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Systems of Linear Differential Equa- tions I. Converting a linear equation to a linear system Consider the third-order equation y ±±± + p ( t ) y ±± + q ( t ) y ± + r ( t ) y = f ( t ) or y ±±± = - r ( t ) y - q ( t ) y ± - p ( t ) y ±± + f ( t ) . 1
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Introduce new dependent variables x 1 ,x 2 3 , as follows: x 1 = y x 2 = x ± 1 (= y ± ) x 3 = x ± 2 (= y ±± ) Then y ±±± = x ± 3 = - r ( t ) x 1 - q ( t ) x 2 - p ( t ) x 3 + f ( t ) The third-order equation can be writ- ten equivalently as the system of three Frst-order equations: 2
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x ± 1 = x 2 x ± 2 = x 3 x ± 3 = - r ( t ) x 1 - q ( t ) x 2 - p ( t ) x 3 + f ( t ) . Note that this system is just a very special case of the “general” sys- tem of three, Frst-order differential equations: 3
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x ± 1 = a 11 ( t ) x 1 + a 12 ( t ) x 2 + a 13 ( t ) x 3 ( t )+ b 1 ( t ) x ± 2 = a 21 ( t ) x 1 + a 22 ( t ) x 2 + a 23 ( t ) x 3 ( t b 2 ( t ) x ± 3 = a 31 ( t ) x 1 + a 32 ( t ) x 2 + a 33 ( t ) x 3 ( t b 3 ( t ) . 4
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Example 1: y ±±± - 3 y ±± - 4 y ± +12 y =3 e t . or y ±±± = - 12 y +4 y ± +3 y ±± e t . Set x 1 = y, x ± 1 = x 2 (= y ± ) ,x ± 2 = x 3 (= y ±± ) . Then y ±±± = x ± 3 = - 12 x 1 x 2 x 3 e t 5
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Equivalent system: x ± 1 = x 2 x ± 2 = x 3 x ± 3 = - 12 x 1 +4 x 2 +3 x 3 e t . II. General Linear Systems Let a 11 ( t ) ,a 12 ( t ) , ..., a nn ( t ) ,b 1 ( t ) 2 ( t ) , .. . be continuous functions on the in- terval I . The system of n Frst- order differential equations 6
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x ± 1 = a 11 ( t ) x 1 + a 12 ( t ) x 2 + ··· + a 1 n ( t ) x n + b 1 ( t ) x ± 2 = a 21 ( t ) x 1 + a 22 ( t ) x 2 + + a 2 n ( t ) x n + b 2 ( t ) . . . . . . x ± n = a n 1 ( t ) x 1 + a n 2 ( t ) x 2 + + a nn ( t ) x n + b n ( t ) is called a frst-order linear differential sys- tem . The system is homogeneous if b 1 ( t ) b 2 ( t ) ≡···≡ b n ( t ) 0o n I. It is nonhomogeneous if the func- tions b i ( t ) are not all identically zero on I . 7
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Set A ( t )= a 11 ( t ) a 12 ( t ) ··· a 1 n ( t ) a 21 ( t ) a 22 ( t ) a 2 n ( t ) . . . . . . . . . a n 1 ( t ) a n 2 ( t ) a nn ( t ) and x = x 1 x 2 . . . x n , b ( t b 1 ( t ) b 2 ( t ) . . . b n ( t ) . The system can be written in the vector-matrix form x ± = A ( t ) x + b ( t ) . (S) 8
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The matrix A ( t ) is called the ma- trix of coefficients or the coeffi- cient matrix . Example 2: y ±±± - 3 y ±± - 4 y ± +12 y =3 e t . Set x 1 = y, x ± 1 = x 2 (= y ± ) ,x ± 2 = x 3 (= y ±± ) . 9
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Equivalent system: x ± 1 = x 2 x ± 2 = x 3 x ± 3 = - 12 x 1 +4 x 2 +3 x 3 e t . Vector-matrix form: x 1 x 2 x 3 ± = 010 001 - 12 4 3 x 1 x 2 x 3 + 0 0 3 e t 10
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A solution of the linear differential system (S) is a differentiable vector function v = x 1 ( t ) x 2 ( t ) . . . x n ( t ) that satisFes (S) on the interval I . Example 3: y ±±± - 3 y ±± - 4 y ± +12 y =3 e t . ±undamental set: ± y 1 = e 3 t ,y 2 = e 2 t 3 = e - 2 t ² 11
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Particular solution: z = 1 2 e t Corresponding solutions of the sys- tem: x 1 = e 3 t 3 e 3 t 9 e 3 t + e t / 2 e t / 2 e t / 2 , x 2 =?
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This note was uploaded on 02/09/2011 for the course MATH 3321 taught by Professor Morgan during the Fall '08 term at University of Houston.

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Chapter 6 Slides - Systems of Linear Dierential Equations I Converting a linear equation to a linear system Consider the third-order equation y or

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