Explanation Given: { e x , e − x , e 2 x } is a fundamental solution set for the homogeneous equation corresponding to the equation y ‴ − 2 y ″ − y ′ + 2 y = g ( x ) . Approach: a) First write down the complimentary function of the given equation. b) Then find out four Wronskian of determinants. c) Formula for particular solution of the equation is y p ( x ) = ∑ k = 1 n y k ( x ) ∫ g ( x ) W K ( x ) W [ y 1 , ... , y n ] ( x ) d x Calculation: To determine the particular solution of the equation first write down the complimentary function by considering fundamental set of solution, C F = c 1 e x + c 2 e − x + c 3 x e 2 x y c = c 1 e x + c 2 e − x + c 3 x e 2 x Compare with C F = c 1 y 1 ( x ) + c 2 y 2 ( x ) + c 3 y 3 ( x ) . Now y 1 ( x ) = e x , y 2 ( x ) = e − x and y 3 ( x ) = x e 2 x Write four Wronskian of determinant, W [ y 1 , y 2 , y 3 ] ( x ) = | y 1 y 2 y 3 y ′ 1 y ′ 2 y ′ 3 y ″ 1 y ″ 2 y ″ 3 | = | e x e − x e 2 x e x − e − x 2 e 2 x e x e − x 4 e 2 x | Now solve and apply operation R 3 = R 3 − R 1 , W [ y 1 , y 2 , y 3 ] ( x ) = | e x e − x e 2 x e x − e − x 2 e 2 x 0 0 3 e 2 x | = 3 e 2 x ( − e x e − x − e − x e x ) W [ y 1 , y 2 , y 3 ] ( x ) = − 6 e 2 x Now find the value of W 1 , W 2 and W 3 W 1 ( x ) = ( − 1 ) 3 − 1 W [ y 2 , y 3 , ] ( x ) = ( − 1 ) 2 | e x x 2 e x ( x + 1 ) e x ( x 2 + 2 x ) e x | Write common factor from each of the column. W 1 ( x ) = ( − 1 ) 3 − 1 W [ y 2 , y 3 , ] ( x ) = | e − x e 2 x − e − x 2 e 2 x | = ( 2 e x + e x ) W 1 ( x ) = 3 e x Similarly for W 2 follow the same procedure.

7th Edition

ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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 6.4, Problem 9E

To determine

The particular solution of the equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.4, Problem 8E

Chapter 6.4, Problem 10E

Students have asked these similar questions

Diagonalize the following matrix, if possible. 8 0 6 - 8 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 8 0 OA. For P= D= 0 3 6 0 B. For P = D= 0 -6 8 0 C. For P = D= 0 - 8 D. The matrix cannot be diagonalized.

Calculus lll May I please have the solutions for the following exercises? Thank you

Calculus lll May I please have the solution for the following question? Thank you

Chapter 6 Solutions

MYLAB MATH-W/ETEXT F/FUND.DIFF.EQUAT.

Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems7-14, determine whether the given...Ch. 6.1 - In Problems7-14, determine whether the given...Ch. 6.1 - In Problems7-14, determine whether the given...

Knowledge Booster

Learn more about

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

The particular solution of the equation.

Solution Summary: The author explains how to determine the particular solution of the given equation.

Concept explainers

Videos

The particular solution of the equation.

Want to see the full answer?

Chapter 6 Solutions