Find the eigenvalues i, and eigenfunctions y,(x) for the given boundary-value problem. (Give your ans y" + Ày = 0, y(0) = 0, y'(T/2) = 0 2, = (2n – 1)2 n = 1, 2, 3, ... Y,(x) = sin (2n – 1)x ,n = 1, 2, 3, ...
Find the eigenvalues i, and eigenfunctions y,(x) for the given boundary-value problem. (Give your ans y" + Ày = 0, y(0) = 0, y'(T/2) = 0 2, = (2n – 1)2 n = 1, 2, 3, ... Y,(x) = sin (2n – 1)x ,n = 1, 2, 3, ...
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Find the eigenvalues \( \lambda_n \) and eigenfunctions \( y_n(x) \) for the given boundary-value problem. (Give your answers in terms of \( n \), making sure that each value of \( n \) corresponds to a unique eigenvalue.)
\[
y'' + \lambda y = 0, \quad y(0) = 0, \quad y(\pi/2) = 0
\]
\[
\lambda_n = \left(2n - 1\right)^2, \quad n = 1, 2, 3, \ldots
\]
✔
\[
y_n(x) = \sin\left((2n - 1)x\right), \quad n = 1, 2, 3, \ldots
\]
✘
The figure presents a mathematical boundary-value problem where you are tasked to identify the eigenvalues and eigenfunctions given the conditions \( y(0) = 0 \) and \( y(\pi/2) = 0 \). The correct eigenvalues and form of the corresponding eigenfunctions are provided for values of \( n \) starting from 1 upwards.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c39d812-58a1-4630-9d2c-3fab5b52adb8%2F41ba09ef-51df-4839-b818-ef3712d37848%2Fiz8yj7_processed.png&w=3840&q=75)
Transcribed Image Text:Find the eigenvalues \( \lambda_n \) and eigenfunctions \( y_n(x) \) for the given boundary-value problem. (Give your answers in terms of \( n \), making sure that each value of \( n \) corresponds to a unique eigenvalue.)
\[
y'' + \lambda y = 0, \quad y(0) = 0, \quad y(\pi/2) = 0
\]
\[
\lambda_n = \left(2n - 1\right)^2, \quad n = 1, 2, 3, \ldots
\]
✔
\[
y_n(x) = \sin\left((2n - 1)x\right), \quad n = 1, 2, 3, \ldots
\]
✘
The figure presents a mathematical boundary-value problem where you are tasked to identify the eigenvalues and eigenfunctions given the conditions \( y(0) = 0 \) and \( y(\pi/2) = 0 \). The correct eigenvalues and form of the corresponding eigenfunctions are provided for values of \( n \) starting from 1 upwards.
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