Verly that each of Umn (x14₁3) = e satisfies Laplace's equation Uxx+lay +lzz = 0 and the boundary Conditions Uco, y, z) = U(1₁ 4₁ 3) = 0, Show i twat N 11 (x₁4₁3) = 2 U₁ (x14₁3) Azl Satisfies the same any -3√m² +n² è as my sunnix m=0₁1₁2₁ Uly (2₁ 0₁ 8) = Uly (2₁ 17, 3) = 0 linear combination n=1₁2₁--- differential equation and boundary conditions.
Verly that each of Umn (x14₁3) = e satisfies Laplace's equation Uxx+lay +lzz = 0 and the boundary Conditions Uco, y, z) = U(1₁ 4₁ 3) = 0, Show i twat N 11 (x₁4₁3) = 2 U₁ (x14₁3) Azl Satisfies the same any -3√m² +n² è as my sunnix m=0₁1₁2₁ Uly (2₁ 0₁ 8) = Uly (2₁ 17, 3) = 0 linear combination n=1₁2₁--- differential equation and boundary conditions.
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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![The problem is to verify that each of the functions
\[ U_{mn}(x,y,z) = e^{3 \sqrt{m^2 + n^2} \, \overline{z}} \cos my \sin nx \quad m = 0, 1, 2, \ldots, \quad n = 1, 2, \ldots \]
satisfies Laplace's equation
\[ U_{xx} + U_{yy} + U_{zz} = 0 \]
and the boundary conditions
\[ U(0, y, z) = U(\pi, y, z) = 0, \]
\[ U_y(x, 0, z) = U_y(x, \pi, z) = 0 \]
Show that any linear combination
\[ U(x, y, z) = \sum_{n=1}^{N} U_n(x, y, z) \]
satisfies the same differential equation and boundary conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F326d8dc8-a191-4bb3-b5b4-0d65333825bc%2Fcb8ufy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The problem is to verify that each of the functions
\[ U_{mn}(x,y,z) = e^{3 \sqrt{m^2 + n^2} \, \overline{z}} \cos my \sin nx \quad m = 0, 1, 2, \ldots, \quad n = 1, 2, \ldots \]
satisfies Laplace's equation
\[ U_{xx} + U_{yy} + U_{zz} = 0 \]
and the boundary conditions
\[ U(0, y, z) = U(\pi, y, z) = 0, \]
\[ U_y(x, 0, z) = U_y(x, \pi, z) = 0 \]
Show that any linear combination
\[ U(x, y, z) = \sum_{n=1}^{N} U_n(x, y, z) \]
satisfies the same differential equation and boundary conditions.
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