Problem 2. In the solution to the problem Uses + Иду U₂ (014) = 0, Ux (1₁4) = 0 11 (71₁0) = 1-X for OLXLI, 450 1 U(my) is bounded as y 500 Find the coefficient on los (6x) 2². er ско C120, leads to trivial solution, You can slap this case). solution: 00 (1(x) = A + Σ An CB(2nx) e-zny 미디 iT/2 So¹¹²2 (1-xc) cs6ze dac A3 A₂ = // Jo NIE 2
Problem 2. In the solution to the problem Uses + Иду U₂ (014) = 0, Ux (1₁4) = 0 11 (71₁0) = 1-X for OLXLI, 450 1 U(my) is bounded as y 500 Find the coefficient on los (6x) 2². er ско C120, leads to trivial solution, You can slap this case). solution: 00 (1(x) = A + Σ An CB(2nx) e-zny 미디 iT/2 So¹¹²2 (1-xc) cs6ze dac A3 A₂ = // Jo NIE 2
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
![### Problem 2
In the solution to the problem
\[ U_{xx} + U_{yy} = 0 \]
for \( 0 < x < \frac{\pi}{2}, y > 0 \)
\[ U_x(0,y) = 0, \quad U_x\left(\frac{\pi}{2}, y\right) = 0 \]
\[ U(x,0) = 1-x, \quad U(x,y) \text{ is bounded as } y \to \infty \]
Find the coefficient on \( \cos(6x) e^{-6y} \). (\( \lambda < 0 \) leads to trivial solution, you can skip this case.)
#### Solution:
\[ U(x,y) = \frac{A_0}{2} + \sum_{n=1}^{\infty} A_n \cos(2nx) e^{-2ny} \]
\[ A_3 = \frac{2}{\pi/2} \int_0^{\pi/2} (1-x) \cos(6x) \, dx = \frac{2}{9\pi} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F03303378-0b20-41bb-b231-4182ad5dccae%2Fpk29akr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 2
In the solution to the problem
\[ U_{xx} + U_{yy} = 0 \]
for \( 0 < x < \frac{\pi}{2}, y > 0 \)
\[ U_x(0,y) = 0, \quad U_x\left(\frac{\pi}{2}, y\right) = 0 \]
\[ U(x,0) = 1-x, \quad U(x,y) \text{ is bounded as } y \to \infty \]
Find the coefficient on \( \cos(6x) e^{-6y} \). (\( \lambda < 0 \) leads to trivial solution, you can skip this case.)
#### Solution:
\[ U(x,y) = \frac{A_0}{2} + \sum_{n=1}^{\infty} A_n \cos(2nx) e^{-2ny} \]
\[ A_3 = \frac{2}{\pi/2} \int_0^{\pi/2} (1-x) \cos(6x) \, dx = \frac{2}{9\pi} \]
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