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x1 = -x2, x2 = 6x₁5x2 with x₁ (0) = 2 and x₂ (0) = 5. (O) Dand (0)

x1 = -x2, x2 = 6x₁5x2 with x₁ (0) = 2 and x₂ (0) = 5. (O) Dand (0)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Do b and d for 6.2.1 Please write 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-. alued 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 = 6x15x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x₁ = x1-x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. · DEC ● (d) x₁ = -2x13x2, 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 = -9x1 + 2x2 +8x3 with x₁ (0) = -2, x2 (0) = 2, and x3 (0) = -4. (g) x₁ = 4x₁-x2+2x3x4, x2 = x1 x3 + x4, x3 = -x3, x4 = x1-x2-2x4 with x₁ (0) = 2, x2 (0) = 1, x3 (0) = 4, and x4 (0) = 1. 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 tv IZA
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-. alued 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 = 6x15x2 with x₁ (0) = 2 and x2 (0) = 5. (c) x₁ = x1-x2, x2 = 5x1 - 3x2 with x₁ (0) = 0 and x₂ (0) = 2. · DEC ● (d) x₁ = -2x13x2, 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 = -9x1 + 2x2 +8x3 with x₁ (0) = -2, x2 (0) = 2, and x3 (0) = -4. (g) x₁ = 4x₁-x2+2x3x4, x2 = x1 x3 + x4, x3 = -x3, x4 = x1-x2-2x4 with x₁ (0) = 2, x2 (0) = 1, x3 (0) = 4, and x4 (0) = 1. 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 tv IZA
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