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

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations

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Chapter 5.2, Problem 25E

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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.

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Chapter 5.2, Problem 24E

Chapter 5.2, Problem 27E

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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...

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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.

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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.

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Chapter 5 Solutions