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Find a function u(x, y) which satisfies the Laplace equation 1 = Urr + = U₁ + 7/124606 = 0 Au in the disc x² + y² <6, and which satisfies the data u(x, y) = y + y² on the disc's boundary. Express your answer in terms of Cartesian coordinates.

Find a function u(x, y) which satisfies the Laplace equation 1 = Urr + = U₁ + 7/124606 = 0 Au in the disc x² + y² <6, and which satisfies the data u(x, y) = y + y² on the disc's boundary. Express your answer in terms of Cartesian coordinates.

Advanced Engineering Mathematics
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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5) Find a function u(x, y) which satisfies the Laplace equation 1 1 Au = Urr + ur +2400 = 0 in the disc x² + y² <6, and which satisfies the data u(x, y) = y + y² on the disc's boundary. Express your answer in terms of Cartesian coordinates. Hint: transform the boundary data, and solve the Laplacian, in terms of polar coordi- nates. Once the particular solution is found, transform back to Cartesian coordinates. The identity 2 sin² 0 = 1- cos(20) may be useful.
Transcribed Image Text:5) Find a function u(x, y) which satisfies the Laplace equation 1 1 Au = Urr + ur +2400 = 0 in the disc x² + y² <6, and which satisfies the data u(x, y) = y + y² on the disc's boundary. Express your answer in terms of Cartesian coordinates. Hint: transform the boundary data, and solve the Laplacian, in terms of polar coordi- nates. Once the particular solution is found, transform back to Cartesian coordinates. The identity 2 sin² 0 = 1- cos(20) may be useful.
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