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y== Cosinlrx) a Ca Cosla.) aSatustied if x=

y== Cosinlrx) a Ca Cosla.) aSatustied if x=

Advanced Engineering Mathematics
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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I don't understand why alpha=(2n-1)*pi/2 and not alpha=n*pi. Can you please explain it to me. Thank you 

The handwritten notes explore a mathematical analysis involving differential equations and eigenfunctions, focusing on boundary conditions and parameter dependencies. --- ### Notes Transcription **Case: \( x < 0 \) → \( y(x) = 0 \)** **Case: \( \lambda = \alpha^2 > 0 \), \( \alpha > 0 \)** - \( y(x) = C_1 \cos(\alpha x) + C_2 \sin(\alpha x) \) - \( 0 = y(0) = C_1 \) - → \( y(x) = C \sin(\alpha x) \) (boxed in red) - \( 0 = y'(1) = \alpha C_2 \cos(\alpha \cdot 1) \) - **Satisfied if** \( \alpha = \frac{(2n-1)\pi}{2} \) → **Eigenvalues of x:** - \( \lambda_n = \alpha_n^2 = \left(\frac{(2n-1)\pi}{2}\right)^2 \) - \( n \in \mathbb{N} \) **Eigenfunctions:** - \( y_n(x) = \sin(\alpha_n x) \) - \( = \sin\left(\frac{(2n-1)\pi x}{2}\right) \) --- ### Explanation - **Boundary Conditions:** - The solution \( y(x) = 0 \) when \( x < 0 \). - The solution considers conditions at \( x = 0 \) and \( x = 1 \) to determine constants and forms. - **Eigenvalues and Eigenfunctions:** - Eigenvalues are calculated as the square of \(\alpha\), determined by boundary conditions. - Eigenfunctions are sine functions dependent on \(\alpha\), showing oscillatory behavior. These notes illustrate solving boundary-value problems typical in physics and engineering contexts, focusing on trigonometric solutions.
Transcribed Image Text:The handwritten notes explore a mathematical analysis involving differential equations and eigenfunctions, focusing on boundary conditions and parameter dependencies. --- ### Notes Transcription **Case: \( x < 0 \) → \( y(x) = 0 \)** **Case: \( \lambda = \alpha^2 > 0 \), \( \alpha > 0 \)** - \( y(x) = C_1 \cos(\alpha x) + C_2 \sin(\alpha x) \) - \( 0 = y(0) = C_1 \) - → \( y(x) = C \sin(\alpha x) \) (boxed in red) - \( 0 = y'(1) = \alpha C_2 \cos(\alpha \cdot 1) \) - **Satisfied if** \( \alpha = \frac{(2n-1)\pi}{2} \) → **Eigenvalues of x:** - \( \lambda_n = \alpha_n^2 = \left(\frac{(2n-1)\pi}{2}\right)^2 \) - \( n \in \mathbb{N} \) **Eigenfunctions:** - \( y_n(x) = \sin(\alpha_n x) \) - \( = \sin\left(\frac{(2n-1)\pi x}{2}\right) \) --- ### Explanation - **Boundary Conditions:** - The solution \( y(x) = 0 \) when \( x < 0 \). - The solution considers conditions at \( x = 0 \) and \( x = 1 \) to determine constants and forms. - **Eigenvalues and Eigenfunctions:** - Eigenvalues are calculated as the square of \(\alpha\), determined by boundary conditions. - Eigenfunctions are sine functions dependent on \(\alpha\), showing oscillatory behavior. These notes illustrate solving boundary-value problems typical in physics and engineering contexts, focusing on trigonometric solutions.
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