Explanation Given: The given system is x ′ = − 6 x + 9 y − ln t , y ′ = − 4 x + 6 y + ln t . Procedure used: First find the eigenvalues using | A − λ I | = 0 . A solution of λ to this equation gives eigenvalues and vector v satisfying ( A − λ I ) v = 0 is an eigenvectors corresponds to λ . From the eigenvalues and eigenvectors, X c ( t ) = c 1 v 1 e λ 1 t + c 2 v 2 e λ 2 t + ⋯ + c n v n e λ n t . The particular solution for the system is X p ( t ) = Φ ( t ) ∫ Φ − 1 ( t ) F ( t ) d t . Here, Φ ( t ) is the fundamental matrix. The general solution is, X ( t ) = X c + X p . Calculation: Consider the system x ′ = − 6 x + 9 y − ln t , y ′ = − 4 x + 6 y + ln t . Convert into matrix form: ( x y ) ′ = ( − 6 9 − 4 6 ) ( x y ) + ( − ln t ln t ) Apply the formula to find the eigenvalues: | A − λ I | = | ( − 6 9 − 4 6 ) − λ ( 1 0 0 1 ) | = | ( − 6 9 − 4 6 ) − ( λ 0 0 λ ) | = | ( − 6 − λ 9 − 4 6 − λ ) | = ( − 6 − λ ) ( 6 − λ ) + 36 Further simplify as: ( − 6 − λ ) ( 6 − λ ) + 36 = − ( 6 + λ ) ( 6 − λ ) + 36 = − ( 36 − λ 2 ) + 36 = λ 2 − 36 + 36 = λ 2 Equate characteristic polynomial to zero in order to calculate λ : λ 2 = 0 λ = 0 The eigenvalues are λ 1 , 2 = 0 . The eigenvector corresponds to λ 1 = 0 : | A − λ I | = ( − 6 9 − 4 6 ) − ( 0 ) ( 1 0 0 1 ) = ( − 6 9 − 4 6 ) − ( 0 0 0 0 ) = ( − 6 9 − 4 6 ) Further reduce as: ( − 6 9 − 4 6 ) → R 2 → R 2 − 2 3 R 1 ( − 6 9 0 0 ) It indicate that, − 6 x 1 + 9 y 1 = 0 . Choose x 1 = 1 , then y 1 = 2 3 gives v 1 = ( 1 2 3 ) . The solution X 1 is: X 1 = ( 1 2 3 ) e 0 t = ( 1 2 3 ) Consider the eigenvector w = ( x 2 y 2 ) : ( − 6 9 − 4 6 ) ( x 2 y 2 ) = ( 1 2 3 ) Can be reducing as: ( − 6 9 − 4 6 | 1 2 3 ) → R 2 → R 2 − 2 3 R 1 ( − 6 9 0 0 | 1 0 ) It indicate that, − 6 x 2 + 9 y 2 = 1 . Choose x 2 = 1 , then y 2 = 7 9 gives w 2 = ( 1 7 9 )

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Differential Equations

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1. Check to see if the matrix below is (a) symmetric (b) skew-symmetric (c) orthogonal You need to check all three cases. 2. Compute the eigenvalues and verify that they satisfy the required properties for such matrix. (For example, if the matrix is symmetric, then the eigenvalues satisfy certain property. Likewise, if a matrix is skew-symmetric or if it is orthogonal.) 0.28 0.96 -0.96 0.28

14. (a) Find all the eigenvalues and their algebraic and geometric multiplicities. (b) Find bases for the corresponding eigenspaces. (c) Is the matrix is diagonalizable? Explain. If it is, then find P and D. 5 2 -9 6

on donne f(x) da fonction derive dhe do fonction fcsos calcule f'(x) orans chacun des Cas sulants: 3 1) f(x)=5x-11, 2- f (x) = ->³ 3-1(x) = x² 12x +π; 4-f(x)=- 5-f(x) = 33-4x6-609)=-3x²+ 7= f(x) = x + 1.8-f(x) = 4 s-f(x) = x++ X+1 -x-1 2 I 3x-4 дево

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Introductory Differential Equations

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