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• Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use (6.31) (with (6.30)) to find a general solution.

• Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use (6.31) (with (6.30)) to find a general solution.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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6.2.2 Please write out Do parts c and d
Exercise 6.2.2 The systems in (a)-(d) involve defective matrices. For each system: Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use (6.31) (with (6.30)) to find a general solution. • Use the general solution to obtain the given initial data. (a) x₁ = 3x₁ - x2, *2 = 4x1 -x2 with x₁ (0) = 1 and x₂ (0) = 3. (b) x₁ = 7x13x2, x2 = 12x15x2 with x₁ (0) = 1 and x₂ (0) = 1. (c) x₁ = -10x₁8x2, x2 = 8x1 + 6x2 with x₁ (0) = 2 and x₂ (0) = 0. (d) x₁ = 6x₁ +5x2+4x3, x2 = -2x1x2x3, x3 = -6x₁-6x2-5x3 with x₁ (0)=1, x₂ (0) = 2, and x3(0) = -1. (This is a 3 x 3 system, but the technique of Section 6.2.4 can easily be adapted.) ● Exercise 6.2.3 Consider a spring-mass system governed by mx"(t) + cx' (t) +kx(t) = 0. (a) Let m = 1, c = 3, and k = 2. Write out the characteristic equation for this ODE, find the roots, and explicitly write out a general solution. (b) Again with m = 1, c = 3, and k = 2, convert this second-order ODE into a pair of coupled first-order ODEs; use x₁ = x and x2 = x. Formulate this system as x = Ax by writing out Nal A DEC 13 tv
Transcribed Image Text:Exercise 6.2.2 The systems in (a)-(d) involve defective matrices. For each system: Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use (6.31) (with (6.30)) to find a general solution. • Use the general solution to obtain the given initial data. (a) x₁ = 3x₁ - x2, *2 = 4x1 -x2 with x₁ (0) = 1 and x₂ (0) = 3. (b) x₁ = 7x13x2, x2 = 12x15x2 with x₁ (0) = 1 and x₂ (0) = 1. (c) x₁ = -10x₁8x2, x2 = 8x1 + 6x2 with x₁ (0) = 2 and x₂ (0) = 0. (d) x₁ = 6x₁ +5x2+4x3, x2 = -2x1x2x3, x3 = -6x₁-6x2-5x3 with x₁ (0)=1, x₂ (0) = 2, and x3(0) = -1. (This is a 3 x 3 system, but the technique of Section 6.2.4 can easily be adapted.) ● Exercise 6.2.3 Consider a spring-mass system governed by mx"(t) + cx' (t) +kx(t) = 0. (a) Let m = 1, c = 3, and k = 2. Write out the characteristic equation for this ODE, find the roots, and explicitly write out a general solution. (b) Again with m = 1, c = 3, and k = 2, convert this second-order ODE into a pair of coupled first-order ODEs; use x₁ = x and x2 = x. Formulate this system as x = Ax by writing out Nal A DEC 13 tv
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