of • verify that each oth functions Ub (ory) = y₁ Un (xry) = nunh my conx Sater fier Laplac's equation Uxx + Uyy = 0 осхип, окуля and the three boundary conclutions. 0, U₂₁ (0₁ y) = U₂ (TT₁ y) = 0₁ 1(²₁0) =0. linear conctition combination show that n=1, 2₁. any U(my) = Aoy + Σ An sinh ny asnxc n=1 Satis furs the same differential equation and boundary conditions.
of • verify that each oth functions Ub (ory) = y₁ Un (xry) = nunh my conx Sater fier Laplac's equation Uxx + Uyy = 0 осхип, окуля and the three boundary conclutions. 0, U₂₁ (0₁ y) = U₂ (TT₁ y) = 0₁ 1(²₁0) =0. linear conctition combination show that n=1, 2₁. any U(my) = Aoy + Σ An sinh ny asnxc n=1 Satis furs the same 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
Related questions
Question
0011
Please show all the steps
![**Verify that each of the functions**
\[
U_0(x,y) = y
\]
\[
U_n(x,y) = \sinh n y \cos n x
\]
\[
n = 1, 2, \ldots
\]
**satisfies the Laplace's equation**
\[
U_{xx} + U_{yy} = 0
\]
for
\[
0 < x < \pi, \quad 0 < y < 2
\]
**and the three boundary conditions**
\[
U_x(0,y) = U_x(\pi,y) = 0, \quad U(x,0) = 0
\]
**Show that any linear combination**
\[
U(x,y) = A_0 y + \sum_{n=1}^{N} A_n \sinh n y \cos n x
\]
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%2F0fa4da6b-f9a1-4eb1-9fb0-f47a9f73c4ce%2F2f3p4bdi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Verify that each of the functions**
\[
U_0(x,y) = y
\]
\[
U_n(x,y) = \sinh n y \cos n x
\]
\[
n = 1, 2, \ldots
\]
**satisfies the Laplace's equation**
\[
U_{xx} + U_{yy} = 0
\]
for
\[
0 < x < \pi, \quad 0 < y < 2
\]
**and the three boundary conditions**
\[
U_x(0,y) = U_x(\pi,y) = 0, \quad U(x,0) = 0
\]
**Show that any linear combination**
\[
U(x,y) = A_0 y + \sum_{n=1}^{N} A_n \sinh n y \cos n x
\]
satisfies the same differential equation and boundary conditions.
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