To determine The general solution of the differential equation. Explanation Consider the differential equation y ‴ − y ″ + y ′ − y = e − t sin t . 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 associated with first column as follows. W 1 ( y 1 , y 2 , y 3 ) ( t ) = | 0 y 2 y 3 0 y ′ 2 y ′ 3 e − t sin t y ″ 2 y ″ 3 | = | 0 cos t sin t 0 − sin t cos t e − t sin t − cos t − sin t | = cos t ( e − t sin t cos t ) + sin t ( e − t sin 2 t ) = e t sin t Similarly, the latter determinant by minors associated with second column as follows. W 2 ( y 1 , y 2 , y 3 ) ( t ) = | y 1 0 y 3 y ′ 1 0 y ′ 3 y ″ 1 e − t sin t y ″ 3 | = | e t 0 sin t e t 0 cos t e t e − t sin t − sin t | = e t ( − e − t sin t cos t ) + sin t ( sin t ) = sin 2 t − sin t cos t The latter determinant by minors associated with third column as follows

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 5P

To determine

The general solution of the differential equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 4P

Chapter 4.4, Problem 6P

Students have asked these similar questions

ax+b proof that se = - è (e" -1)" ë nax

20.11 ← UAS Sisa waktu 01:20:01 51%- Soal 2 Perhatikan gambar di bawah (Sembunyikan ) Belum dijawab Ditandai dari 1,00 5 A B E D 10 20 Jika ruas garis AB, PE, dan DC sejajar dan ketiganya tegak lurus dengan ruas garis BC, maka panjang ruas garis PE adalah ... (cukup tulis bilangannya tanpa spasi dalam bentuk desimal tiga angka di belakang koma, seperti a,bcd atau pecahan m/n untuk m n Jawaban: Jawaban ||| <

20.07 52% X https://www.chegg.com/hc <: C Chegg Learn on the go = Chegg (X) Open in app EN-US ✔ What's your next question? √x #16 A surveyor sees a building across the river. Standing at point A he measures the angle of elevation from the ground to the top of the building to be 30 degrees. He steps back 100 feet and again measures the angle of elevation and finds it to be 15. (See Figure 12.26.) Assuming that it makes a 90-degree angle with the floor, approximately how tall is the building? 15 30° 100 A river Figure 12.26 Show image transcript Here's the best way to solve it. Solution ||| о building < Share

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.