Explanation Consider the given systems of differential equation as { y ′ 1 = 10 y 1 − 10 y 2 − 4 y 3 y ′ 2 = − 10 y 1 + y 2 − 14 y 3 y ′ 3 = − 4 y 1 − 14 y 2 − 2 y 3 . The coefficient matrix of the corresponding systems of differential equations is A = [ 10 − 10 − 4 − 10 1 − 14 − 4 − 14 − 2 ] . Note that, a single vector equation is y ′ = A y , where A is coefficient matrix . Hence, a single vector equation is y ′ = [ 10 − 10 − 4 − 10 1 − 14 − 4 − 14 − 2 ] y , where y = [ y 1 y 2 y 3 ] . Characteristic equation of the coefficient matrix A = [ 10 − 10 − 4 − 10 1 − 14 − 4 − 14 − 2 ] is det ( A − λ I ) = 0 , where λ eigenvalue and I is 3 × 3 identity matrix . Simplify for the det ( A − λ I ) = 0 using the coefficient matrix as follows. det ( A − λ I ) = 0 det ( [ 10 − 10 − 4 − 10 1 − 14 − 4 − 14 − 2 ] − λ [ 1 0 0 0 1 0 0 0 1 ] ) = 0 det ( [ 10 − λ − 10 − 4 − 10 1 − λ − 14 − 4 − 14 − 2 − λ ] ) = 0 ( 10 − λ ) { ( 1 − λ ) ( − 2 − λ ) − 196 } + 10 { − 10 ( − 2 − λ ) − 56 } − 4 { 140 + 4 ( 1 − λ ) } = 0 Further simplified as, ( 10 − λ ) { λ 2 + λ − 198 } + 10 { 20 + 10 λ − 56 } − 4 { 140 + 4 − 4 λ } = 0 ( 10 − λ ) { λ 2 + λ − 198 } + 100 λ − 360 − 576 + 16 λ = 0 λ 2 ( λ − 9 ) − 324 ( λ − 9 ) = 0 ( λ − 9 ) ( 324 − λ 2 ) = 0 λ − 9 = 0 or 324 − λ 2 = 0 λ = 9 or λ = ± 18 Compute the eigenvector x ( 1 ) = [ x 1 x 2 x 3 ] corresponding to the eigenvalue λ = 9 as follows. A x − ( − 9 ) x = 0 [ 10 − λ − 10 − 4 − 10 1 − λ − 14 − 4 − 14 − 2 − λ ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ ( 10 − λ ) x 1 − 10 x 2 − 4 x 3 − 10 x 1 + ( 1 − λ ) x 2 − 14 x 3 − 4 x 1 − 14 x 2 − ( 2 + λ ) x 3 ] = [ 0 0 0 ] ( 10 − λ ) x 1 − 10 x 2 − 4 x 3 = 0 , − 10 x 1 + ( 1 − λ ) x 2 − 14 x 3 = 0 and − 4 x 1 − 14 x 2 − ( 2 + λ ) x 3 = 0 Consider the general eigenvector equation for eigenvalue λ as ( 10 − λ ) x 1 − 10 x 2 − 4 x 3 = 0

10th Edition

ISBN: 9781119809210

Author: Kreyszig

Publisher: WILEY

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Chapter 4.3, Problem 9P

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The real general solution of the given systems of differential equation.

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Chapter 4.3, Problem 8P

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ADV.ENG.MATH (LL) W/WILEYPLUS BUNDLE

Ch. 4.1 - Prob. 1P Ch. 4.1 - Prob. 2P Ch. 4.1 - Prob. 3P Ch. 4.1 - Prob. 4P Ch. 4.1 - If you extend Example 1 by a tank T3 of the same...Ch. 4.1 - Find a “general solution” of the system in Prob....Ch. 4.1 - In Example 2 find the currents: 7. If the initial...Ch. 4.1 - Prob. 8P Ch. 4.1 - Prob. 9P Ch. 4.1 - Find a general solution of the given ODE (a) by...

Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Find a general solution of the given ODE (a) by...Ch. 4.1 - Prob. 14P Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - 1–9 GENERAL SOLUTION Find a real general solution...Ch. 4.3 - Find a real general solution of the following...Ch. 4.3 - Prob. 9P Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - 10–15 IVPs Solve the following initial value...Ch. 4.3 - Prob. 12P Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Solve the following initial value problems. Ch. 4.3 - Prob. 16P Ch. 4.3 - Prob. 17P Ch. 4.3 - Prob. 18P Ch. 4.3 - Prob. 19P Ch. 4.4 - Prob. 1P Ch. 4.4 - Prob. 2P Ch. 4.4 - Prob. 3P Ch. 4.4 - Prob. 4P Ch. 4.4 - Prob. 5P Ch. 4.4 - Prob. 6P Ch. 4.4 - Prob. 7P Ch. 4.4 - Prob. 8P Ch. 4.4 - Prob. 9P Ch. 4.4 - Prob. 10P Ch. 4.4 - Prob. 11P Ch. 4.4 - Prob. 12P Ch. 4.4 - Prob. 13P Ch. 4.4 - Prob. 14P Ch. 4.4 - Prob. 15P Ch. 4.4 - Prob. 16P Ch. 4.4 - Prob. 17P Ch. 4.5 - Prob. 1P Ch. 4.5 - Prob. 2P Ch. 4.5 - Prob. 4P Ch. 4.5 - Prob. 5P Ch. 4.5 - Prob. 6P Ch. 4.5 - Prob. 7P Ch. 4.5 - Prob. 8P Ch. 4.5 - Prob. 9P Ch. 4.5 - Prob. 10P Ch. 4.5 - Prob. 11P Ch. 4.5 - Prob. 12P Ch. 4.5 - Prob. 13P Ch. 4.6 - Prob. 1P Ch. 4.6 - Prob. 2P Ch. 4.6 - Prob. 3P Ch. 4.6 - Prob. 4P Ch. 4.6 - Prob. 5P Ch. 4.6 - Prob. 6P Ch. 4.6 - Prob. 7P Ch. 4.6 - Prob. 9P Ch. 4.6 - Prob. 10P Ch. 4.6 - Prob. 11P Ch. 4.6 - Prob. 12P Ch. 4.6 - Prob. 13P Ch. 4.6 - Prob. 14P Ch. 4.6 - Prob. 15P Ch. 4.6 - Prob. 16P Ch. 4.6 - Prob. 17P Ch. 4.6 - Prob. 19P Ch. 4 - Prob. 1RQ Ch. 4 - Prob. 2RQ Ch. 4 - How can you transform an ODE into a system of...Ch. 4 - What are qualitative methods for systems? Why are...Ch. 4 - Prob. 5RQ Ch. 4 - Prob. 6RQ Ch. 4 - What are eigenvalues? What role did they play in...Ch. 4 - Prob. 8RQ Ch. 4 - Prob. 9RQ Ch. 4 - Prob. 10RQ Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Find a general solution. Determine the kind and...Ch. 4 - Prob. 15RQ Ch. 4 - Prob. 16RQ Ch. 4 - Prob. 17RQ Ch. 4 - Prob. 18RQ Ch. 4 - Prob. 19RQ Ch. 4 - Prob. 20RQ Ch. 4 - Prob. 21RQ Ch. 4 - Prob. 22RQ Ch. 4 - Prob. 23RQ Ch. 4 - Prob. 24RQ Ch. 4 - Prob. 25RQ Ch. 4 - Network. Find the currents in Fig. 103 when R = 1...Ch. 4 - Prob. 27RQ Ch. 4 - Prob. 28RQ Ch. 4 - Find the location and kind of all critical points...Ch. 4 - Find the location and kind of all critical points...

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