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ди 10²u = It 40x²¹ together with the boundary conditions u(0, t) = 0 and u(, t) = 0 for t≥0 and the initial condition u(x, 0) = x(x) for 0 < x (a) If n is a positive integer, show that the function Un(x, t) = e-n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is 8 u(x,t) = Σ Bhe e-n²t sin(2nx). n=1 Wnito d /1

ди 10²u = It 40x²¹ together with the boundary conditions u(0, t) = 0 and u(, t) = 0 for t≥0 and the initial condition u(x, 0) = x(x) for 0 < x (a) If n is a positive integer, show that the function Un(x, t) = e-n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is 8 u(x,t) = Σ Bhe e-n²t sin(2nx). n=1 Wnito d /1

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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Consider the partial differential equation Ju 10² u Ət 40x²¹ together with the boundary conditions u(0, t) = 0 and u(,t) = 0 for t≥0 and the initial condition u(x, 0) = x(x) for 0 < x </ (a) If n is a positive integer, show that the function Un(x, t) = e¯n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is 8 u(x, t)=B₂e-n²t sin(2nx). n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn.
Transcribed Image Text:Consider the partial differential equation Ju 10² u Ət 40x²¹ together with the boundary conditions u(0, t) = 0 and u(,t) = 0 for t≥0 and the initial condition u(x, 0) = x(x) for 0 < x </ (a) If n is a positive integer, show that the function Un(x, t) = e¯n²t sin(2nx), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is 8 u(x, t)=B₂e-n²t sin(2nx). n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn.
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