To determine The general solution of the higher order differential equation y ( 8 ) + 8 y ( 4 ) + 16 y = 0 . Explanation Procedure used: General solution of a higher order differential equation. 1. In general, to solve a n th order differential equation a n y n + a n − 1 y n − 1 + ⋯ + a 2 y ″ + a 1 ′ + a 0 y = 0 , it is required to find the solution of a n th order polynomial equation a n r n + a n − 1 r n − 1 + ⋯ + a 0 = 0 . 2. If all the roots of a n r n + a n − 1 r n − 1 + ⋯ + a 0 = 0 are real and distinct, then the general solution will be y = c 1 e r 1 x + c 2 e r 2 x + ... + c n e r n x . 3. If an auxiliary equation has k roots and all the roots of the auxiliary equation are equal, consider that all the roots are equal to r 1 then all the solutions are linearly independent. In such case the general solution of the higher order differential equation must contain the linear combination c 1 e r 1 x + c 2 x e r 1 x + c 3 x 2 e r 1 x + … + c k x k − 1 e r 1 x . 4. In case if higher order differential equation has repeated conjugate complex roots. Then the general solution of the higher order differential equation will be, y = c 1 e r 1 x + c 2 e − r 1 x + c 3 x e r 1 x + c 4 x e − r 1 x where r 1 represents an imaginary root. Calculation: Consider the given differential equation y ( 8 ) + 8 y ( 4 ) + 16 y = 0 . Obtain the characteristic equation as follows. r 8 + 8 r 4 + 16 = 0 ( r 4 + 4 ) 2 = 0 r 4 + 4 = 0 ( r 2 ) 2 − ( 2 i ) 2 = 0 On further simplification, ( r 2 + 2 i ) ( r 2 − 2 i ) = 0 ( r 2 + 2 i ) = 0 or ( r 2 − 2 i ) = 0 r 2 = − 2 i or r 2 − 2 i = 0 ( a + b i ) 2 = − 2 i or ( a + b i ) 2 = 2 i a 2 − b 2 + 2 i a b = 0 − 2 i or a 2 − b 2 + 2 i a b = 0 + 2 i Equate the real and complex number as follows. a 2 − b 2 = 0 ⋯ ( 1 ) 2 a b = − 2 ⋯ ( 2 ) } and a 2 − b 2 = 0 ⋯ ( 3 ) 2 a b = 2 ⋯ ( 4 ) } From the equation (2), 2 a b = − 2 a = − 1 b Substitute a = − 1 b in equation (1)

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 4.2, Problem 21P

To determine

The general solution of the higher order differential equation y(8)+8y(4)+16y=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.2, Problem 20P

Chapter 4.2, Problem 22P

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

Elementary Differential Equations

Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - In each of Problems 1 through 6, determine...Ch. 4.1 - Prob. 6P Ch. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 8P Ch. 4.1 - In each of Problems 7 through 10, determine...Ch. 4.1 - Prob. 10P

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.