9. Show that the following functions satisfy the equation Urr +Uyy = 0, known as the two-dimensional Laplace equation: (a) u(x,y) = x² − y² − 3xy + 5x − 6 (b) u(x, y) = ln(x² + y²), (x,y) ‡ (0,0) (c) u(x, y) = eª sin y. 10. Show that the following functions satisfy the equation Utt = Uxx + Uyy, known as the two-space-dimensional wave equation: (a) u(x, y, t) = e¹-t — 3e³+¹ +7xy - t + 13 (b) u(x, y, t) = (3x − 4y + 5t)ª, a a constant.
9. Show that the following functions satisfy the equation Urr +Uyy = 0, known as the two-dimensional Laplace equation: (a) u(x,y) = x² − y² − 3xy + 5x − 6 (b) u(x, y) = ln(x² + y²), (x,y) ‡ (0,0) (c) u(x, y) = eª sin y. 10. Show that the following functions satisfy the equation Utt = Uxx + Uyy, known as the two-space-dimensional wave equation: (a) u(x, y, t) = e¹-t — 3e³+¹ +7xy - t + 13 (b) u(x, y, t) = (3x − 4y + 5t)ª, a a constant.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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do 9b 10a
Transcribed Image Text:9. Show that the following functions satisfy the equation Urr +Uyy = 0, known
as the two-dimensional Laplace equation:
(a) u(x,y) = x² − y² − 3xy + 5x − 6
(b) u(x, y) = ln(x² + y²), (x,y) ‡ (0,0)
(c) u(x, y) = eª sin y.
10. Show that the following functions satisfy the equation Utt = Uxx + Uyy,
known as the two-space-dimensional wave equation:
(a) u(x, y, t) = e¹-t — 3e³+¹ +7xy - t + 13
(b) u(x, y, t) = (3x − 4y + 5t)ª, a a constant.
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