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Problem #1: (a) Consider the Laplace equation શક Oy² = 0, x² +1² <1, (1) with the boundary condition v(x, y) = 11x²y² + 813 on the circle x² + y² = 1. (2) Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t). (b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (1) and satisfies the boundary condition (2). Enter v(x, y).

Problem #1: (a) Consider the Laplace equation શક Oy² = 0, x² +1² <1, (1) with the boundary condition v(x, y) = 11x²y² + 813 on the circle x² + y² = 1. (2) Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t). (b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (1) and satisfies the boundary condition (2). Enter v(x, y).

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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Solve a and b part only I need perfect soloution plz And 100 perent correct soloution. I need this answer in maximum 2 hours so plz don't take tooo much and solve this and take a thumb up i will give u rating also so plz solve
Problem #1: (a) Consider the Laplace equation Ox² Oy² = 0, x² +1² <1, (1) with the boundary condition v(x, y) = 11x²y² + 813 on the circle x² + y² = 1. ox (2) Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t). (b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (1) and satisfies the boundary condition (2). Enter v(x, y). (c) Consider the Laplace equation + = 0, ₁²2² +1² > 1, (3) with the boundary condition (2). Use polar coordinates to solve (3) (i.e. consider the function u(r, 8) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded near infinity, i.e. that lim u(r, 0)| < 0. Replacing the variable by t, enter the function u(r, t). (d) Express the solution found in (c) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (3) and satisfies the boundary condition (2). Enter v(x, y). Hint: Use standard trigonometric formulas to write the initial condition in a more convenient form when solving (a) and (c).
Transcribed Image Text:Problem #1: (a) Consider the Laplace equation Ox² Oy² = 0, x² +1² <1, (1) with the boundary condition v(x, y) = 11x²y² + 813 on the circle x² + y² = 1. ox (2) Use polar coordinates to solve (1) (i.e. consider the function u(r, 0) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded at the origin, i.e., u(0, 0) <. Replacing the variable by t, enter the function u(r, t). (b) Express the solution found in (a) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (1) and satisfies the boundary condition (2). Enter v(x, y). (c) Consider the Laplace equation + = 0, ₁²2² +1² > 1, (3) with the boundary condition (2). Use polar coordinates to solve (3) (i.e. consider the function u(r, 8) = v(r cos(8), r sin(0)). You may assume that u(r, 0) is bounded near infinity, i.e. that lim u(r, 0)| < 0. Replacing the variable by t, enter the function u(r, t). (d) Express the solution found in (c) in rectangular coordinates, i.e. compute the function v(x, y) that is a solution of (3) and satisfies the boundary condition (2). Enter v(x, y). Hint: Use standard trigonometric formulas to write the initial condition in a more convenient form when solving (a) and (c).
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