×
Log in to StudySoup
Get Full Access to Elementary Differential Equations And Boundary Value Problems - 10 Edition - Chapter 4.1 - Problem 20
Join StudySoup for FREE
Get Full Access to Elementary Differential Equations And Boundary Value Problems - 10 Edition - Chapter 4.1 - Problem 20

Already have an account? Login here
×
Reset your password

In this problem we show how to generalize Theorem 3.2.7

Elementary Differential Equations and Boundary Value Problems | 10th Edition | ISBN: 9780470458310 | Authors: William E. Boyce ISBN: 9780470458310 168

Solution for problem 20 Chapter 4.1

Elementary Differential Equations and Boundary Value Problems | 10th Edition

  • Textbook Solutions
  • 2901 Step-by-step solutions solved by professors and subject experts
  • Get 24/7 help from StudySoup virtual teaching assistants
Elementary Differential Equations and Boundary Value Problems | 10th Edition | ISBN: 9780470458310 | Authors: William E. Boyce

Elementary Differential Equations and Boundary Value Problems | 10th Edition

4 5 1 342 Reviews
11
4
Problem 20

In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higherorder equations. We first outline the procedure for the third order equationy+ p1(t)y+ p2(t)y+ p3(t)y = 0.Let y1, y2, and y3 be solutions of this equation on an interval I.(a) If W = W(y1, y2, y3), show thatW=y1 y2 y3y1 y2 y3y1 y2 y3.Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinantsobtained by differentiating the first, second, and third rows, respectively.(b) Substitute for y1 , y2 , and y3 from the differential equation; multiply the first row byp3, multiply the second row by p2, and add these to the last row to obtainW= p1(t)W.(c) Show thatW(y1, y2, y3)(t) = c exp p1(t) dt.It follows that W is either always zero or nowhere zero on I.(d) Generalize this argument to the nth order equationy(n) + p1(t)y(n1) ++ pn(t)y = 0with solutions y1, ... , yn. That is, establish Abels formulaW(y1, ... , yn)(t) = c exp p1(t) dtfor this case.

Step-by-Step Solution:
Step 1 of 3

Derivatives: Derivative is a special kindof limit. It’s a,it’s a kind of limit. The definition of the derivative can be approached in two different ways.One is geometrical (as a slopeof a curve) and the other one is physical (as a rate of change). A derivative at a point is the slope of the line which is tangent to the function at point . Function  Interval  [ ,] The Average rate of change in the interval , is given by: − ( ) − The rate of change is the slope of the line going through point [, ] , [, ]. Definition: The derivative of a function is denoted by:

Step 2 of 3

Chapter 4.1, Problem 20 is Solved
Step 3 of 3

Textbook: Elementary Differential Equations and Boundary Value Problems
Edition: 10
Author: William E. Boyce
ISBN: 9780470458310

The answer to “In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higherorder equations. We first outline the procedure for the third order equationy+ p1(t)y+ p2(t)y+ p3(t)y = 0.Let y1, y2, and y3 be solutions of this equation on an interval I.(a) If W = W(y1, y2, y3), show thatW=y1 y2 y3y1 y2 y3y1 y2 y3.Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinantsobtained by differentiating the first, second, and third rows, respectively.(b) Substitute for y1 , y2 , and y3 from the differential equation; multiply the first row byp3, multiply the second row by p2, and add these to the last row to obtainW= p1(t)W.(c) Show thatW(y1, y2, y3)(t) = c exp p1(t) dt.It follows that W is either always zero or nowhere zero on I.(d) Generalize this argument to the nth order equationy(n) + p1(t)y(n1) ++ pn(t)y = 0with solutions y1, ... , yn. That is, establish Abels formulaW(y1, ... , yn)(t) = c exp p1(t) dtfor this case.” is broken down into a number of easy to follow steps, and 167 words. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Since the solution to 20 from 4.1 chapter was answered, more than 588 students have viewed the full step-by-step answer. This full solution covers the following key subjects: . This expansive textbook survival guide covers 76 chapters, and 2039 solutions. The full step-by-step solution to problem: 20 from chapter: 4.1 was answered by , our top Math solution expert on 12/23/17, 04:36PM.

Other solutions

People also purchased

Related chapters

Unlock Textbook Solution

Enter your email below to unlock your verified solution to:

In this problem we show how to generalize Theorem 3.2.7