To determine The real general solution of the given systems of differential equation. Explanation Consider the given systems of differential equation as { y ′ 1 = y 2 y ′ 2 = − y 1 + y 3 y ′ 3 = − y 2 . The coefficient matrix of the corresponding systems of differential equations is A = [ 0 1 0 − 1 0 1 0 − 1 0 ] . Note that, a single vector equation is y ′ = A y , where A is coefficient matrix . Hence, a single vector equation is y ′ = [ 0 1 0 − 1 0 1 0 − 1 0 ] y , where y = [ y 1 y 2 y 3 ] . Characteristic equation of the coefficient matrix A = [ 0 1 0 − 1 0 1 0 − 1 0 ] 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 ( [ 0 1 0 − 1 0 1 0 − 1 0 ] − λ [ 1 0 0 0 1 0 0 0 1 ] ) = 0 det ( [ 0 − λ 1 0 − 1 0 − λ 1 0 − 1 0 − λ ] ) = 0 − λ 3 − 2 λ = 0 Further simplified as, λ ( λ 2 + 1 ) = 0 λ = 0 or λ 2 + 2 = 0 λ = 0 or λ = ± 2 i Compute the eigenvector x ( 1 ) = [ x 1 x 2 x 3 ] corresponding to the eigenvalue λ = 0 as follows. A x − ( 0 ) x = 0 [ 0 − 0 1 0 − 1 0 − 0 1 0 − 1 0 − 0 ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ x 2 − x 1 + x 3 x 2 ] = [ 0 0 0 ] x 2 = 0 , x 2 =0 and − x 1 + x 3 = 0 x 2 = 0 and x 1 1 = x 3 1 Hence, the possible vector corresponding to the eigenvalue λ = 0 is x ( 1 ) = [ 1 0 1 ] . Compute the eigenvector x ( 2 ) = [ x 1 x 2 x 3 ] corresponding to the eigenvalue λ = − 2 i as follows. A x − ( − 2 i ) x = 0 [ 0 + 2 i 1 0 − 1 0 + 2 i 1 0 − 1 0 + 2 i ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ 2 i x 1 + x 2 − x 1 + 2 i x 2 + x 3 − x 2 + 2 i x 3 ] = [ 0 0 0 ] 2 i x 1 + x 2 = 0 , − x 1 + 2 i x 2 + x 3 =0 and − x 2 + 2 i x 3 = 0 x 1 − 1 = x 2 2 i and x 2 2 i = x 3 1 Hence, the possible vector corresponding to the eigenvalue λ = − 2 i is x ( 2 ) = [ − 1 2 i 1 ] . Compute the eigenvector x ( 2 ) = [ x 1 x 2 x 3 ] corresponding to the eigenvalue λ = 2 i as follows. A x − ( 2 i ) x = 0 [ 0 − 2 i 1 0 − 1 0 − 2 i 1 0 − 1 0 − 2 i ] [ x 1 x 2 x 3 ] = [ 0 0 0 ] [ − 2 i x 1 + x 2 − x 1 − 2 i x 2 + x 3 − x 2 − 2 i x 3 ] = [ 0 0 0 ] − 2 i x 1 + x 2 = 0 , − x 1 − 2 i x 2 + x 3 =0 and − x 2 − 2 i x 3 = 0 x 1 1 = x 2 2 i and x 2 2 i = x 3 − 1 Hence, the possible vector corresponding to the eigenvalue λ = i is x ( 3 ) = [ 1 2 i − 1 ]

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Question

Chapter 4.3, Problem 7P

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.3, Problem 6P

Chapter 4.3, Problem 8P

Students have asked these similar questions

12:25 AM Sun Dec 22 uestion 6- Week 8: QuX Assume that a company X + → C ezto.mheducation.com Week 8: Quiz i Saved 6 4 points Help Save & Exit Submit Assume that a company is considering purchasing a machine for $50,000 that will have a five-year useful life and a $5,000 salvage value. The machine will lower operating costs by $17,000 per year. The company's required rate of return is 15%. The net present value of this investment is closest to: Click here to view Exhibit 12B-1 and Exhibit 12B-2, to determine the appropriate discount factor(s) using the tables provided. 00:33:45 Multiple Choice О $6,984. $11,859. $22,919. ○ $9,469, Mc Graw Hill 2 100-

No chatgpt pls will upvote

7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)

Chapter 4 Solutions

Advanced Engineering Mathematics

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...

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.