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for 2x" + 4x' + 5x = 10e¹ is -t

for 2x" + 4x' + 5x = 10e¹ is -t

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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Question 5-7
5. The state-space representation for 2x" + 4x' + 5x = 10e¹ is [x₂] = - [4] []e X₂ 98 99 X2 0 1 6. Calculate the eigenvalue of the state-space coefficient matrix -7a-2a using the methods demonstrated in your lecture notes (Note that a is a positive constant, do not assume values for a). If your eigenvalues are real and different, let ₁ be the smaller of the two eigenvalues when comparing their absolute values, for example, if your eigenvalues are -3 and -7, their absolute values are 3 and 7 with 3 < 7 and ₁= -3. If your eigenvalues are a complex conjugate pair, let ₁ be the eigenvalue with the positive imaginary part. The eigenvalue you must keep is 2₁ = 9₁1 a + 912 a j Note that if is real valued that 912 = 0 7. The general solution of a non homogeneous state-space equation is given below. Use the initial conditions to determine the value of C₁. [*] = C₁ Ⓡ²¹ [2¹] + C₂ (-²) ¹ [_-2₁ ₁] +0 ;] C₁e(2j) given x(0) = 0, x' (0) = 2 You calculated that C₁ = 913 + 914 j Note that if C₁ is a real number that 914 = 0. +
Transcribed Image Text:5. The state-space representation for 2x" + 4x' + 5x = 10e¹ is [x₂] = - [4] []e X₂ 98 99 X2 0 1 6. Calculate the eigenvalue of the state-space coefficient matrix -7a-2a using the methods demonstrated in your lecture notes (Note that a is a positive constant, do not assume values for a). If your eigenvalues are real and different, let ₁ be the smaller of the two eigenvalues when comparing their absolute values, for example, if your eigenvalues are -3 and -7, their absolute values are 3 and 7 with 3 < 7 and ₁= -3. If your eigenvalues are a complex conjugate pair, let ₁ be the eigenvalue with the positive imaginary part. The eigenvalue you must keep is 2₁ = 9₁1 a + 912 a j Note that if is real valued that 912 = 0 7. The general solution of a non homogeneous state-space equation is given below. Use the initial conditions to determine the value of C₁. [*] = C₁ Ⓡ²¹ [2¹] + C₂ (-²) ¹ [_-2₁ ₁] +0 ;] C₁e(2j) given x(0) = 0, x' (0) = 2 You calculated that C₁ = 913 + 914 j Note that if C₁ is a real number that 914 = 0. +
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