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

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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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…

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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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Differential Equations with Boundary-Value Problems (MindTap Course List)

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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