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a) Let u(x, t) = X(x)T(t) and show that and X" + XX=0, X (0) = 0, X' (L) + yX (L) = 0, T' + Aa²T = 0, where A is the separation constant. (b) Assume that A is real, and show that problem (ii) has no nontrivial solutions if X < 0. (c) If A > 0, let λ = μ² with μ> 0. Show that problem (ii) has nontrivial solutions only if u is a solution of the equation cos μL + y sin μL = 0. (ii)

a) Let u(x, t) = X(x)T(t) and show that and X" + XX=0, X (0) = 0, X' (L) + yX (L) = 0, T' + Aa²T = 0, where A is the separation constant. (b) Assume that A is real, and show that problem (ii) has no nontrivial solutions if X < 0. (c) If A > 0, let λ = μ² with μ> 0. Show that problem (ii) has nontrivial solutions only if u is a solution of the equation cos μL + y sin μL = 0. (ii)

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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25. Consider the problem a²U u(0,t) =0, uc u(x,0) =f(x), 0≤x≤L. = Ut, 0<x<L, t> 0 (L,t)+yu(L,t)=0, t>0 0
Transcribed Image Text:25. Consider the problem a²U u(0,t) =0, uc u(x,0) =f(x), 0≤x≤L. = Ut, 0<x<L, t> 0 (L,t)+yu(L,t)=0, t>0 0
(a) Let u(x, t) = X(x)T(t) and show that and X" + XX=0, X (0) = 0, X' (L) +yX (L) = 0, T' + Aa²T = 0, where A is the separation constant. (b) Assume that A is real, and show that problem (ii) has no nontrivial solutions if X < 0. (c) If À > 0, let À = μ² withμ > 0. Show that problem (ii) has nontrivial solutions only if u is a solution of the equation μcos μL + y sin μL = 0. (ii)
Transcribed Image Text:(a) Let u(x, t) = X(x)T(t) and show that and X" + XX=0, X (0) = 0, X' (L) +yX (L) = 0, T' + Aa²T = 0, where A is the separation constant. (b) Assume that A is real, and show that problem (ii) has no nontrivial solutions if X < 0. (c) If À > 0, let À = μ² withμ > 0. Show that problem (ii) has nontrivial solutions only if u is a solution of the equation μcos μL + y sin μL = 0. (ii)
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