Part Cc please Solve the following problems(a) Find the eigenvalues and eigen-functions for the problemƒ” (x) + \ƒ(x) = 0, ƒ(0) = 0, ƒ'(2) = 0(b) Discuss the orthogonality relation for the eigen-functions in part (a)(c) Find the general solution of the problemut (x, t) =9uxx (x, t), 0≤x≤2, t>0u(0, t)=0, ux (2, t) = 0

(a) f''(x)+λfx=0, f0=0, f'2=0 Let λ=0 fx=ax+b⇒f0=0⇒b=0f'(2)=0⇒a=0 Therefore f(x)=0 For λ=-μ2…

Answered: (a) Find the eigenvalues and…

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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Question

Part Cc please

Solve the following problems
(a) Find the eigenvalues and eigen-functions for the problem
ƒ” (x) + \ƒ(x) = 0, ƒ(0) = 0, ƒ'(2) = 0
(b) Discuss the orthogonality relation for the eigen-functions in part (a)
(c) Find the general solution of the problem
ut (x, t) =9uxx (x, t), 0≤x≤2, t>0
u(0, t)=0, ux (2, t) = 0

Expert Solution

Step 1: Finding the eigen value and eigen function

(a) $f'' (x) + λ f (x) = 0, f (0) = 0, f' (2) = 0$

Let $λ = 0$

$f (x) = a x + b \Rightarrow f (0) = 0 \Rightarrow b = 0 f' (2) = 0 \Rightarrow a = 0$

Therefore $f (x) = 0$

For $λ = - μ^{2}$

$f " (x) - μ^{2} f (x) = 0 f (x) = A e^{μ x} + B e^{- μ x} ∴ f (0) = A + B = 0 \Rightarrow f (x) = A e^{μ x} - A e^{- μ x} \Rightarrow f' (x) = A μ e^{μ x} + A μ e^{- μ x} \Rightarrow f' (2) = A μ e^{μ 2} + A μ e^{- μ 2} = 0 \Rightarrow A = 0 ∴ f (x) = 0$

For $λ = μ^{2}$

$f " (x) + μ^{2} f (x) = 0 f (x) = A \cos (μ x) + B \sin (μ x) ∴ f (0) = A = 0 \Rightarrow f (x) = B \sin (μ x) \Rightarrow f' (x) = B μ \cos (μ x) \Rightarrow f' (2) = B μ \cos (μ 2) = 0 ∴ 2 μ = (n π + \frac{π}{2}) \Rightarrow μ = \frac{2 n π + π}{4}$