2130_Chapter7_Notes

2130_Chapter7_Notes - APMA 2130 Section 7.1 Systems of...

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APMA 2130 Section 7.1: Systems of First Order Linear Equations Post-Lecture Handout Dr. Bernard Fulgham University of Virginia July 23, 2013 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exa 2 (slide 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exa 2 (slide 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Matrix Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1
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Introduction System of n first order ordinary differential equations: x 1 = F 1 ( t, x 1 , x 2 , . . . , x n ) , x 2 = F 2 ( t, x 1 , x 2 , . . . , x n ) , . . . x n = F n ( t, x 1 , x 2 , . . . , x n ) , where each x k is a function of t. If each F k is a linear function of x 1 , . . . , x n , then the system of equations is linear ; otherwise, it is nonlinear . Systems of higher order differential equations can similarly be defined. 2 / 9 Linear Systems Linear system of equations: x 1 = p 11 ( t ) x 1 + p 12 ( t ) x 2 + ··· + p 1 n ( t ) x n + g 1 ( t ) , x 2 = p 21 ( t ) x 1 + p 22 ( t ) x 2 + + p 2 n ( t ) x n + g 2 ( t ) , . . . x n = p n 1 ( t ) x 1 + p n 2 ( t ) x 2 + + p nn ( t ) x n + g n ( t ) . If each g k ( t ) = 0 , then the system is homogeneous ; otherwise, it is nonhomogeneous . Every n th order linear equation can be converted to a system of n first order linear equations. 3 / 9 2
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Example 1 Third order linear equation: y ′′′ + 2 y ′′ + 3 y + 4 y = g ( t ) . Define x 1 = y, x 2 = y , x 3 = y ′′ , then x 1 = y = x 1 = x 2 , x 2 = y ′′ = x 2 = x 3 , x 3 = y ′′′ = - 4 y - 3 y - 2 y ′′ + g ( t ) = x 3 = - 4 x 1 - 3 x 2 - 2 x 3 + g ( t ) . System of three first order linear equations: x 1 = x 2 , x 2 = x 3 , x 3 = - 4 x 1 - 3 x 2 - 2 x 3 + g ( t ) . 4 / 9 Theorem Linear system of n first order equations: x 1 = p 11 ( t ) x 1 + p 12 ( t ) x 2 + ··· + p 1 n ( t ) x n + g 1 ( t ) , x 2 = p 21 ( t ) x 1 + p 22 ( t ) x 2 + + p 2 n ( t ) x n + g 2 ( t ) , . . . x n = p n 1 ( t ) x 1 + p n 2 ( t ) x 2 + + p nn ( t ) x n + g n ( t ) . Initial conditions: x 1 ( t 0 ) = x 0 1 , x 2 ( t 0 ) = x 0 2 , . . . , x n ( t 0 ) = x 0 n . Questions When does a solution exist? If a solution exists, is it unique? Theorem 7.1.2 If the coefficient functions p 11 ( t ) , p 12 ( t ) , . . . , p nn ( t ) , g 1 ( t ) , . . . , g n ( t ) are continuous on an open interval I = ( α, β ) , then the system with initial conditions has a unique solution x 1 = φ 1 ( t ) , x 2 = φ 2 ( t ) , . . . , x n = φ n ( t ) throughout the entire interval ( α, β ) . 5 / 9 3
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Example 2 x 1 = 3 x 1 - 2 x 2 , x 1 (0) = 3 x 2 = 2 x 1 - 2 x 2 , x 2 (0) = 1 / 2 Solve this system by converting to a second order linear equation: x 1 = 3 x 1 - 2 x 2 = x 2 = 1 2 (3 x 1 - x 1 ) = x 2 = 1 2 (3 x 1 - x ′′ 1 ) . Now substitute for x 2 and x 2 in the second equation: x 2 = 2 x 1 - 2 x 2 = 1 2 (3 x 1 - x ′′ 1 ) = 2 x 1 - (3 x 1 - x 1 ) = 3 x 1 - x ′′ 1 = 4 x 1 - 6 x 1 + 2 x 1 = x ′′ 1 - x 1 - 2 x 1 = 0 . Note: the original system was homogeneous, and so the resulting equation is also homogeneous. 6 / 9 Exa 2 (slide 2) x 1 = 3 x 1 - 2 x 2 x 2 = 2 x 1 - 2 x 2 = x ′′ 1 - x 1 - 2 x 1 = 0 Characteristic equation: r 2 - r - 2 = 0 = ( r - 2)( r + 1) = 0 = r = 2 , - 1 = x 1 = c 1 e 2 t + c 2 e t = x 1 = 2 c 1 e 2 t - c 2 e t , and x 2 = 1 2 (3 x 1 - x 1 ) = 1 2 b 3( c 1 e 2 t + c 2 e t ) - (2 c 1 e 2 t - c 2 e t ) B = 1 2 ( 3 c 1 e 2 t + 3 c 2 e t - 2 c 1 e 2 t + c 2 e t ) = 1 2 c 1 e 2 t + 2 c 2 e t .
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2130_Chapter7_Notes - APMA 2130 Section 7.1 Systems of...

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