A) Classify the partial differential equation 2uxx - 4uxy - 2Uyy +3ux + 4Uy + 6u = 0. as hyperbolic, parabolic or elliptic.
A) Classify the partial differential equation 2uxx - 4uxy - 2Uyy +3ux + 4Uy + 6u = 0. as hyperbolic, parabolic or elliptic.
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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Please answer all parts correctly
![**A)** Classify the partial differential equation
\(2u_{xx} - 4u_{xy} - 2u_{yy} + 3u_x + 4u_y + 6u = 0\).
as hyperbolic, parabolic, or elliptic.
---
**B)** The Fourier cosine representation of the function on \(0 < x < \pi\) is as follows:
\[x^2 \approx \frac{\pi^2}{3} + 4 \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos nx\]
Use the above result to establish the correspondence in the interval \(0 < x < 3\).
---
**C)** Compute the Laplacian of the function
\(u(x, y) = \ln(x^2 + y^2)\)
in an appropriate coordinate system and decide if the given function satisfies Laplace’s equation \(\nabla^2 u = 0\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F5124e560-f416-46be-af57-70e39a943f32%2Fcrh43go_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**A)** Classify the partial differential equation
\(2u_{xx} - 4u_{xy} - 2u_{yy} + 3u_x + 4u_y + 6u = 0\).
as hyperbolic, parabolic, or elliptic.
---
**B)** The Fourier cosine representation of the function on \(0 < x < \pi\) is as follows:
\[x^2 \approx \frac{\pi^2}{3} + 4 \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos nx\]
Use the above result to establish the correspondence in the interval \(0 < x < 3\).
---
**C)** Compute the Laplacian of the function
\(u(x, y) = \ln(x^2 + y^2)\)
in an appropriate coordinate system and decide if the given function satisfies Laplace’s equation \(\nabla^2 u = 0\).
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