Explanation Approach: To solve the given homogeneous differential solution, do the following. Solve the auxiliary equation for the homogeneous differential equations From Newton’s method numerical technique is applied as follows: r n + 1 = r n − f ( r n ) f ′ ( r n ) r 1 = r 0 − f ( r 0 ) f ′ ( r 0 ) Use various iterations to calculate roots for the auxiliary equation. r n + 1 = r n − f ( r n ) f ′ ( r n ) r 1 = r 0 − f ( r 0 ) f ′ ( r 0 ) Change the values r 0 and find the values or r n to find the root of the equation. General solution is obtained with the help of roots. Calculation: Consider D = d d x in the given equation. y ( 4 ) + 2 y ‴ − 3 y ″ − y ′ + 1 2 y = 0 ( D 4 + 2 D 3 − 3 D 2 − D + 1 2 ) y = 0 Auxiliary equation of the above homogeneous equation is as follows. f ( r ) = 0 r 4 + 2 r 3 − 3 r 2 − r + 1 2 = 0 Differentiate above equation with respect to r f ′ ( r ) = 4 r 3 + 6 r 2 − 6 r − 1 (From Newton’s method) r n + 1 = r n − f ( r n ) f ′ ( r n ) r 1 = r 0 − f ( r 0 ) f ′ ( r 0 ) For, r 0 = 0 r 0 = 1 r 0 = − 1 r 0 = − 3 Take r 0 = 0 f ( r 0 ) = r 0 4 + 2 r 0 3 − 3 r 0 2 − r 0 + 1 2 f ′ ( r 0 ) = 4 r 0 3 + 6 r 0 2 − 6 r 0 − 1 r 1 = r 0 − r 0 4 + 2 r 0 3 − 3 r 0 2 − r 0 + 1 2 4 r 0 3 + 6 r 0 2 − 6 r 0 − 1 = 0 + 1 2 = 0.5 r 2 = r 1 − f ( r 1 ) f ′ ( r 1 ) Take r 1 = 0.5 r 2 = r 1 − f ( r 1 ) f ′ ( r 1 ) = r 1 − r 1 4 + 2 r 1 3 − 3 r 1 2 − r 1 + 1 2 4 r 1 3 + 6 r 1 2 − 6 r 1 − 1 = 0.5 − 0.21875 = 0.28125 Similarly, calculate value of r 3 , r 4 , r 5 , r 6 , r 7 , r 8 , and r 9 . r [ 1 ] 0.498505481066269 r [ 2 ] 0.281573887091018 r [ 3 ] 0.296399737129932 r [ 4 ] 0.296324805937571 r [ 5 ] 0.296324780002779 r [ 6 ] 0.296324779994422 r [ 7 ] 0.296324779994419 r [ 8 ] 0.296324779994419 r [ 9 ] 0.296324779994419 r 4 , r 5 , r 6 , r 7 , r 8 , and r 9 values are approximately equal. Therefore, root is 0.296324 Similarly, For r 0 = 1 r 1 = r 0 − f ( r 0 ) f ′ ( r 0 ) f ( r 0 ) = r 0 4 + 2 r 0 3 − 3 r 0 2 − r 0 + 1 2 f ′ ( r 0 ) = 4 r 0 3 + 6 r 0 2 − 6 r 0 − 1 This gives, r 1 = r 0 − r 0 4 + 2 r 0 3 − 3 r 0 2 − r 0 + 1 2 4 r 0 3 + 6 r 0 2 − 6 r 0 − 1 Calculate the values r 1 , r 2 , r 3 , r 4 , r 5 , r 6 , r 7 , r 8 , and r 9 r [ 1 ] 1.16616783078326 r [ 2 ]

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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 6.2, Problem 27E

To determine

To find:

The general solution for the differential equation y(4)+2y‴−3y″−y′+12y=0 using numerical technique.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.2, Problem 26E

Chapter 6.2, Problem 28E

Students have asked these similar questions

25.2. Find the Laurent series for the function 1/[z(z-1)] in the follow- ing domains: (a). 0<|z|< 1, (b). 1<|z, (c). 0

25.5. Find the Laurent series for the function 1/[(z - 1)(-2)(z - 3)] in the following domains: (a). 0 3. شهری

25.1. Expand each of the following functions f(z) in a Laurent series on the indicated domain: (a). z² - 2z+5 (2-2)(z² + 1)' (c). Log za 2 b (z - موجود 11, 29, where b>a> 1 are real, |z| > b.

Chapter 6 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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 general solution for the differential equation y ( 4 ) + 2 y ‴ − 3 y ″ − y ′ + 1 2 y = 0 using numerical technique.

Solution Summary: The author explains how Newton's method solves the homogeneous differential equation using numerical technique.

Concept explainers

Videos

To find:

The general solution for the differential equation y(4)+2y‴−3y″−y′+12y=0 using numerical technique.

Want to see the full answer?

Chapter 6 Solutions

The general solution for the differential equation y ( 4 ) + 2 y ‴ − 3 y ″ − y ′ + 1 2 y = 0 using numerical technique.

Solution Summary: The author explains how Newton's method solves the homogeneous differential equation using numerical technique.

Concept explainers

Videos

To find: The general solution for the differential equation y(4)+2y‴−3y″−y′+12y=0 using numerical technique.

Want to see the full answer?

Chapter 6 Solutions

To find:

The general solution for the differential equation y(4)+2y‴−3y″−y′+12y=0 using numerical technique.