Explanation Given: The given eigenvalue problem is: y ″ + λ y = 0 ; y ( 0 ) = 0 , y ′ ( 1 ) = 0 Procedure used: Find the eigenvalues and eigenfunctions: Consider the equation, a y ″ + b y ′ + c y = q x . First find the roots of the auxiliary equation: a r 2 + b r + c = 0 If the roots are conjugate complex α ± β i , then the general solution form is: y ( x ) = c 1 e α x cos β x + c 2 e α x sin β x If the roots are distinct α 1 , α 2 , ⋯ , then the general solution form is: y ( x ) = c 1 e α 1 x + c 2 e α 2 x + ⋯ If the roots are equal α 1 = α 2 , then the general solution form is: y ( x ) = c 1 x + c 2 After that, use the initial condition to find the value of the constants in the general solution. The function obtained is called eigenfunction and the corresponding value of root is called eigenvalue. Calculation: Consider the eigenvalue problem: y ″ + λ y = 0 ; y ( 0 ) = 0 , y ′ ( 1 ) = 0 The auxiliary equation is: r 2 + λ = 0 r = ± λ i The nature of root varies for λ negative, zero and positive. Case 1: λ < 0 . Let λ = − μ 2 for μ > 0 . Then, the root of the auxiliary equation is: r 2 − μ 2 = 0 r = ± μ The general solution is: y ( x ) = c 1 e μ x + c 2 e − μ x Convert into hyperbolic equation: y ( x ) = C 1 cosh μ x + C 2 sinh μ x Substitute the value zero for x and apply the initial value y ( 0 ) = 0 : y ( 0 ) = C 1 cosh μ ( 0 ) + C 2 sinh μ ( 0 ) 0 = C 1 Differentiate the above equation with respect to x : y ′ ( x ) = C 1 μ sinh μ x + C 2 μ cosh μ x Substitute the value 1 for x in the above equation and apply the initial value y ′ ( 1 ) = 0 : y ′ ( 1 ) = C 1 μ sinh μ ( 1 ) + C 2 μ cosh μ ( 1 ) 0 = C 1 μ sinh μ ( 1 ) + C 2 μ cosh μ ( 1 ) Substitute the value C 1 : C 2 μ cosh μ = 0 To obtain nontrivial solution, we must have C 2 ≠ 0 . Then, cosh μ = 0 , μ = ( n + 1 2 ) π The value of C 1 = 0 and C 2 is any arbitrary value (let’s say c ). There are infinite number of solution with different value of n . so, the constant will be different (let’s say c n ) The solution is: y ( x ) = c n sin ( n + 1 2 ) π x and the eigenvalues are λ n = − μ 2 = − ( n + 1 2 ) 2 π 2

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 11.2, Problem 13E

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To find:

The values of all eigenvalues and eigenfunctions for the given eigenvalue problem y″+λy=0; y(0)=0, y′(1)=0.

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Chapter 11.2, Problem 12E

Chapter 11.2, Problem 14E

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Fundamentals of Differential Equations and Boundary Value Problems

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The values of all eigenvalues and eigenfunctions for the given eigenvalue problem y ″ + λ y = 0 ; y ( 0 ) = 0 , y ′ ( 1 ) = 0 .

Solution Summary: The author explains how to find the eigenvalues of the given auxiliary equation: ar2+br+c.

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To find:

The values of all eigenvalues and eigenfunctions for the given eigenvalue problem y″+λy=0; y(0)=0, y′(1)=0.

Want to see the full answer?

Chapter 11 Solutions

The values of all eigenvalues and eigenfunctions for the given eigenvalue problem y ″ + λ y = 0 ; y ( 0 ) = 0 , y ′ ( 1 ) = 0 .

Solution Summary: The author explains how to find the eigenvalues of the given auxiliary equation: ar2+br+c.

Videos

To find: The values of all eigenvalues and eigenfunctions for the given eigenvalue problem y″+λy=0; y(0)=0, y′(1)=0.

Want to see the full answer?

Chapter 11 Solutions

To find:

The values of all eigenvalues and eigenfunctions for the given eigenvalue problem y″+λy=0; y(0)=0, y′(1)=0.