Explanation Given: The systems of differential equations: d x d t = 5 x − 3 y , d y d t = 4 x − 3 y Approach: 1) The characteristic equation of the matrix M = [ a b c d ] is given by | M − x I | = 0 2) Let M = [ a b c d ] be a matrix with eigenvalue λ . The nonzero vector v such that A v = λ v is called an eigenvector of the matrix M . Calculation: Write the given system in matrix form. d X = [ 5 − 3 4 − 3 ] X . Here, X = [ x y ] Find the eigenvalue of the matrix [ 5 − 3 4 − 3 ] . Eigenvalues are the roots of x 2 − 2 x − 3 = 0 ( x − 3 ) ( x + 1 ) = 0 … ( 1 ) So, the eigenvalues are 3 , − 1 . The eigenvector of the matrix A corresponding to the eigenvalue 3 is given by the solution of the system, [ 5 − 3 − 3 4 − 3 − 3 ] [ x 1 x 2 ] = [ 0 0 ] [ 2 − 3 4 − 6 ] [ x 1 x 2 ] = [ 0 0 ] 2 x 1 − 3 x 2 = 0 x 2 = 2 x 1 3 The solution of the system is [ x 1 x 2 ] = [ x 1 2 x 1 3 ] = [ 1 2 3 ] So, the eigenvector of the matrix A corresponding to the eigenvalue 3 is [ 1 2 3 ]

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Chapter 12.2, Problem 27E

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The eigenvectors of the matrix for the system: dxdt=5x−3y,dydt=4x−3y and show that the eigenvectors are parallel to the lines y=2x3 and y=2x.

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Chapter 12.2, Problem 26E

Chapter 12.3, Problem 1E

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Ch. 12.2 - In Problem 16, classify the critical point at the...Ch. 12.2 - Prob. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Prob. 5E Ch. 12.2 - Prob. 6E Ch. 12.2 - Prob. 7E Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - In Problem 712, find and classify the critical...

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The eigenvectors of the matrix for the system: d x d t = 5 x − 3 y , d y d t = 4 x − 3 y and show that the eigenvectors are parallel to the lines y = 2 x 3 and y = 2 x .

Solution Summary: The author determines the eigenvectors of the matrix for the system, which are parallel to the lines y=2x3 and

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The eigenvectors of the matrix for the system: dxdt=5x−3y,dydt=4x−3y and show that the eigenvectors are parallel to the lines y=2x3 and y=2x.

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Chapter 12 Solutions