Consider the partial differential equation du 182² 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 < 1. (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 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 B- =
Consider the partial differential equation du 182² 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 < 1. (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 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 B- =
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
q34
Transcribed Image Text:Consider the partial differential equation
du
182² 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 < 1.
(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
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 B-
=
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