To determine The general solution of the initial value problem. Explanation Consider the differential equation y ‴ − y ″ + y ′ − y = sec t , y ( 0 ) = 2 , y ′ ( 0 ) = − 1 , y ″ ( 0 ) = 1 . The characteristic polynomial for the equation is r 3 − r 2 + r − 1 . Obtain the root of the equation as follows. r 3 − r 2 + r − 1 = 0 r 2 ( r − 1 ) + ( r − 1 ) = 0 ( r − 1 ) ( r 2 + 1 ) = 0 r = 1 , ± i The roots are r = 1 , ± i . Therefore, the complimentary solution of the equation is y c = c 1 e 1 t + c 2 cos t + c 3 sin t . A fundamental set of solutions y 1 , y 2 and y 3 then, the general solution of the homogeneous equation is y c = c 1 e t + c 2 cos t + c 3 sin t . Solving the fundamental set W ( y 1 , y 2 , y 3 ) ( t ) as follows. W ( y 1 , y 2 , y 3 ) ( t ) = | y 1 y 2 y 3 y ′ 1 y ′ 2 y ′ 3 y ″ 1 y ″ 2 y ″ 3 | = | e t cos t sin t e t − sin t cos t e t − cos t − sin t | = { e t ( sin 2 t + cos 2 t ) − cos t ( − e t sin t − e t cos t ) + sin t ( − e t cos t + e t sin t ) } = 2 e t Compute the latter determinant by minors as follows. W 1 ( y 1 , y 2 , y 3 ) ( t ) = | 0 y 2 y 3 0 y ′ 2 y ′ 3 sec t y ″ 2 y ″ 3 | = | 0 cos t sin t 0 − sin t cos t sec t − cos t − sin t | = − cos t ( − sec t cos t ) + sin t ( sec t sin t ) = cos t + sin t ( sin t cos t ) ( ∵ 1 cos t = sec t ) On further simplification, W 1 ( y 1 , y 2 , y 3 ) ( t ) = cos t + sin 2 t cos t = cos 2 t + sin 2 t cos t ( ∵ cos 2 t + sin 2 t = 1 ) = sec t ( ∵ 1 cos t = sec t ) Similarly, W 2 ( y 1 , y 2 , y 3 ) ( t ) = | y 1 0 y 3 y ′ 1 0 y ′ 3 y ″ 1 sec t y ″ 3 | = | e t 0 sin t e t 0 cos t e t sec t − sin t | = e t ( − sec t cos t ) + sin t ( e t sec t ) = e t ( tan t − 1 ) ( ∵ 1 cos t = sec t , sin t cos t = tan t ) W 3 ( y 1 , y 2 , y 3 ) ( t ) = | y 1 y 2 0 y ′ 1 y ′ 2 0 y ″ 1 y ″ 2 sec t | = | e t cos t 0 e t − sin t 0 e t − cos t sec t | = e t ( − sin t sec t ) − cos t ( e t sec t ) = − e t ( tan t + 1 ) ( ∵ 1 cos t = sec t , sin t cos t = tan t ) By using the variation of parameters formula for obtain the particular solution

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 4.4, Problem 11P

To determine

The general solution of the initial value problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 10P

Chapter 4.4, Problem 12P

Students have asked these similar questions

Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.

3. [10 marks] Let Go (Vo, Eo) and G₁ = (V1, E1) be two graphs that ⚫ have at least 2 vertices each, ⚫are disjoint (i.e., Von V₁ = 0), ⚫ and are both Eulerian. Consider connecting Go and G₁ by adding a set of new edges F, where each new edge has one end in Vo and the other end in V₁. (a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian? (b) If so, what is the size of the smallest possible F? Prove that your answers are correct.

Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.

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.