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In each of Problems 19 through 21, verify that each given function is a solution of the given partial differential equation.

In each of Problems 19 through 21, verify that each given function is a solution of the given partial differential equation.

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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en or ARGE is a Int ich the 0. In each of Problems 16 through 18, determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts. 16. Uxx + Uyy + Uzz = 0 17. Uxxxx +2uxxyy + Uyyyy = 0 www alor 18. ut + uux = 1 + Uxx In each of Problems 19 through 21, verify that each given function is a solution of the given partial differential equation. u₁(x, y) = cos x cosh y, 19. uxx + Uyy = 0; u₂(x, y) = ln(x² + y²) 20. a²uxx = ₁; u₂(x, t) = e-a²x² u₁(x, t) = e-α²₁ sinx, sin Ax, λ a real constant 21. a²uxx = Utti; u₂(x, t) = sin(x - at), A a real constant u₁(x, t) = sin(x) sin(at), 22. Follow the steps indicated here to derive the equation of motion of a pendulum, equation (12) in the text. Assume that the rod is rigid and weightless, that the mass is a point mass, and that there is no friction or drag anywhere in the system. a. Assume that the mass is in an arbitrary displaced position, indicated by the angle 0. Draw a free-body diagram showing the forces acting on the mass. b. Apply Newton's law of motion in the direction tangential to the circular arc on which the mass moves. Then the tensile force in the rod does not enter the equation. Observe that you need to find the component of the gravitational force in the tangential direction. Observe also that the linear acceleration, as opposed to the angular acceleration, is Ld²0/dt2, where L is the length of the rod. c. Simplify the result from part b to obtain equation (12) in the text.
Transcribed Image Text:en or ARGE is a Int ich the 0. In each of Problems 16 through 18, determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts. 16. Uxx + Uyy + Uzz = 0 17. Uxxxx +2uxxyy + Uyyyy = 0 www alor 18. ut + uux = 1 + Uxx In each of Problems 19 through 21, verify that each given function is a solution of the given partial differential equation. u₁(x, y) = cos x cosh y, 19. uxx + Uyy = 0; u₂(x, y) = ln(x² + y²) 20. a²uxx = ₁; u₂(x, t) = e-a²x² u₁(x, t) = e-α²₁ sinx, sin Ax, λ a real constant 21. a²uxx = Utti; u₂(x, t) = sin(x - at), A a real constant u₁(x, t) = sin(x) sin(at), 22. Follow the steps indicated here to derive the equation of motion of a pendulum, equation (12) in the text. Assume that the rod is rigid and weightless, that the mass is a point mass, and that there is no friction or drag anywhere in the system. a. Assume that the mass is in an arbitrary displaced position, indicated by the angle 0. Draw a free-body diagram showing the forces acting on the mass. b. Apply Newton's law of motion in the direction tangential to the circular arc on which the mass moves. Then the tensile force in the rod does not enter the equation. Observe that you need to find the component of the gravitational force in the tangential direction. Observe also that the linear acceleration, as opposed to the angular acceleration, is Ld²0/dt2, where L is the length of the rod. c. Simplify the result from part b to obtain equation (12) in the text.
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