Explanation Property used: (1) If y ″ + λ y = 0 with y ( 0 ) = 0 , y ( L ) = 0 , then for λ > 0 , then write λ = α 2 , where α is a positive number. The solution of the differential equation is y = c 1 cos α x + c 2 sin α x . (2) If y ″ + λ y = 0 with y ( 0 ) = 0 , y ( L ) = 0 , then for λ = 0 , then write y ″ = 0 . The solution of the differential equation is y = c 1 x + c 2 . Calculation: Consider the given differential equation d 2 d x 2 ( E I d 2 y d x 2 ) + P d 2 y d x 2 = 0 . Here, λ = α 2 = P E I . Equation above can be written in simplified format as follows. y ‴ ′ + α 2 y ″ = 0 . The auxiliary equation of the differential equation is m 4 + α 2 m 2 = 0 . Above equation can be simplified as follows. m 2 ( m 2 + α 2 ) = 0 m 2 = 0 or m 2 + α 2 = 0 The roots of the auxiliary equation are m = ± α i and m = ± 0 . There are four roots for the equation. Thus, the solution of the differential equation is, y ( x ) = c 1 cos α x + c 2 sin α x + c 3 x + c 4 . Boundary conditions is used to calculate the values of the constants. The given boundary conditions are y ( 0 ) = 0 , y ″ ( 0 ) = 0 , y ( L ) = 0 , y ″ ( L ) = 0

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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 5.2, Problem 25E

To determine

To prove: That the solution of the given fourth order differential equation subject to the boundary conditions is equivalent to the analysis in Example 4.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.2, Problem 24E

Chapter 5.2, Problem 27E

Students have asked these similar questions

int/PlayerHomework.aspx?homeworkId=689099898&questionId=1&flushed=false&cid=8120746¢erw BP Physical Geograph... HW Score: 0%, 0 of 13 points ○ Points: 0 of 1 Determine if the values of the variables listed are solutions of the system of equations. 2x - y = 4 3x+5y= - 6 x=1, y = 2; (1,-2) Is (1, 2) a solution of the system of equations? L No Yes iew an example Get more help - A

Determine if each statement is true or false. If the statement is false, provide a brief explanation: a) There exists x = R such that √x2 = -x. b) Let A = {x = ZIx = 1 (mod 3)} and B = {x = ZIx is odd}. Then A and B are disjoint. c) Let A and B be subsets of a universal set U. If x = A and x/ € A - B,then x = An B.| E d) Let f : RR be defined by f (x) = 1 x + 2 1. Then f is surjective.

Write the negation of the definition of an injective function

Chapter 5 Solutions

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

Ch. 5.1 - 5.1.1 Spring/Mass systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A force...Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Spring/Mass Systems: Free Undamped Motion A mass...Ch. 5.1 - 5.1.1Spring/Mass Systems: Free Undamped Motion A...

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.

To prove: That the solution of the given fourth order differential equation subject to the boundary conditions is equivalent to the analysis in Example 4.

Solution Summary: The author analyzes the solution of the fourth order differential equation subject to the boundary conditions.

Concept explainers

Videos

To prove: That the solution of the given fourth order differential equation subject to the boundary conditions is equivalent to the analysis in Example 4.

Want to see the full answer?

Chapter 5 Solutions