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Exercise 6.2.1 For each system of ODE in (a)-(g), do the following: • Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use them to write out a general solution using (6.17). ● If any eigenvalues for A are complex, write out a real-valued general solution. • Use either form of the general solution to obtain the given initial data. (a) x₁ = 7x1 - 4x2, x2 = 20x1 - 11x2 with x₁ (0) = 3 and x₂ (0) = 8. (b) x1 = -x2, x2 = 6x15x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x1 = x1 - x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. (d) x₁ = -2x1 - 3x2, x2 = 3x12x2 with x₁ (0) = 2 and x₂(0) = -2. (e) x₁ = -6x1 +9x2 - 4x3, x2 = -6x1+11x2 - 6x3, x3 = -10x1 +21x2 - 12x3 with x₁ (0) = -1, x₂ (0) = 0, and x3 (0) = 2.

Exercise 6.2.1 For each system of ODE in (a)-(g), do the following: • Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use them to write out a general solution using (6.17). ● If any eigenvalues for A are complex, write out a real-valued general solution. • Use either form of the general solution to obtain the given initial data. (a) x₁ = 7x1 - 4x2, x2 = 20x1 - 11x2 with x₁ (0) = 3 and x₂ (0) = 8. (b) x1 = -x2, x2 = 6x15x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x1 = x1 - x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. (d) x₁ = -2x1 - 3x2, x2 = 3x12x2 with x₁ (0) = 2 and x₂(0) = -2. (e) x₁ = -6x1 +9x2 - 4x3, x2 = -6x1+11x2 - 6x3, x3 = -10x1 +21x2 - 12x3 with x₁ (0) = -1, x₂ (0) = 0, and x3 (0) = 2.

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 30E
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Please do e and f for 6.2.1 And write it out
Exercise 6.2.1 For each system of ODE in (a)-(g), do the following: • Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use them to write out a general solution using (6.17). • If any eigenvalues for A are complex, write out a real-valued general solution. ● Use either form of the general solution to obtain the given initial data. (a) x₁ = 7x14x2, x2 = 20x1 - 11x2 with x₁ (0) = 3 and x₂ (0) = 8. (b) x₁ = -x2, x2 = 6x1 - 5x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x1 = x1-x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. DEC 13 (d) x₁ = -2x1 - 3x2, x2 = 3x12x2 with x₁ (0) = 2 and x₂ (0) = -2. (e) x₁ = -6x1 +9x2 - 4x3, x2 = -6x₁+11x2 - 6x3, x3 = -10x1 +21x2 - 12x3 with x₁ (0) = -1, x₂(0) = 0, and x3 (0) = 2. (f) x₁ = -7x₁ + 2x2 +6x3, x2 = -6x1 - x2 + 4x3, x3 x2 (0) = 2, and x3 (0) = -4. (g) x₁ = 4x₁-x2+2x3 x4, x2 = x1 x3 + x4, x3 = x2 (0) = 1, x3 (0) = 4, and x4 (0) = 1. ● -9x1 + 2x2 + 8x3 with x₁ tv == 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 = = -2, -x3, x4 = x1-x2-2x4 with x₁ (0) = 2, Sid A A
Transcribed Image Text:Exercise 6.2.1 For each system of ODE in (a)-(g), do the following: • Formulate the system as x = Ax, by explicitly writing out the matrix A. • Find the eigenvalues and eigenvectors of A and use them to write out a general solution using (6.17). • If any eigenvalues for A are complex, write out a real-valued general solution. ● Use either form of the general solution to obtain the given initial data. (a) x₁ = 7x14x2, x2 = 20x1 - 11x2 with x₁ (0) = 3 and x₂ (0) = 8. (b) x₁ = -x2, x2 = 6x1 - 5x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x1 = x1-x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. DEC 13 (d) x₁ = -2x1 - 3x2, x2 = 3x12x2 with x₁ (0) = 2 and x₂ (0) = -2. (e) x₁ = -6x1 +9x2 - 4x3, x2 = -6x₁+11x2 - 6x3, x3 = -10x1 +21x2 - 12x3 with x₁ (0) = -1, x₂(0) = 0, and x3 (0) = 2. (f) x₁ = -7x₁ + 2x2 +6x3, x2 = -6x1 - x2 + 4x3, x3 x2 (0) = 2, and x3 (0) = -4. (g) x₁ = 4x₁-x2+2x3 x4, x2 = x1 x3 + x4, x3 = x2 (0) = 1, x3 (0) = 4, and x4 (0) = 1. ● -9x1 + 2x2 + 8x3 with x₁ tv == 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 = = -2, -x3, x4 = x1-x2-2x4 with x₁ (0) = 2, Sid A A
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