Explanation Given: The differential equation is, d 2 y d t 2 + 2 c d y d t + k y = 0 Here, c and k are positive constants. Approach: Critical points are the solutions of x ′ = 0 and y ′ = 0 . Eigenvalues are the roots of characteristic equation. Classification of critical points: For complex conjugates eigenvalues: When the real part λ is zero, this type of critical point is called a center. It is neutrally stable. When the real part λ is nonzero, this type of critical point is called a spiral point. It is asymptotically stable if real part is less than 0 , it is unstable if real part is greater than 0 . For real eigenvalues: Distinct eigenavlues When both eigenvalues are positive or both are negative, this type of critical point is called a node. It is asymptotically stable if eigenvalues are both negative, unstable if eigenvalues are both positive. When eigenvalues have opposite signs, this type of critical point is called a saddle point. It is always unstable. Repeated eigenvalues When there are two linearly independent eigenvectors, this type of critical point is called a proper node. It is asymptotically stable if eigenvalue is less than 0 and unstable if eigenvalue is greater than 0 . When there is only one linearly independent eigenvector, this type of critical point is called an improper node. It is asymptotically stable if eigenvalue is less than 0 , unstable if eigenavalue is greater than 0 . Calculation: Consider, d 2 y d t 2 + 2 c d y d t + k y = 0 Let u = y and v = y ′ . Now, d u d t = v d v d t = − k u − 2 c v Equate to zero. Then, d u d t = 0 This implies, v = 0 And, d v d t = 0 k u − 2 c v = 0 Thus, u = 0 The critical point is ( 0 , 0 ) . It can be written in matrix form as, x ′ = [ 0 1 − k − 2 c ] x Obtain the eigenvalues. A = [ 0 1 − k − 2 c ]

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Differential Equations and Linear Algebra (4th Edition)

Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 5TFR Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 7TFR Ch. 9.1 - For Question a-j, decide if the given statement is...Ch. 9.1 - Prob. 9TFR Ch. 9.1 - Prob. 10TFR

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The first order linear system of the differential equation and the corresponding phase portraits.

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