Consider the partial differential equation ди 18²u 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 u(x, t) =B₁e-n²t sin(2nx). Σ n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn. -

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Consider the partial differential equation
ди
18²u
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 < 1/1.
(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
8
u(x, t) =B₁e-n²t sin(2nx).
Σ
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 ди 18²u 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 < 1/1. (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 8 u(x, t) =B₁e-n²t sin(2nx). Σ n=1 Write down (but do not evaluate) an integral that can be used to determine the constants Bn. -
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