The function u(x, y) satisfies the partial differential equation J²u J²u xy- əx² дхду +x². ди Y əx (a) Show that this equation is hyperbolic. (b) Find the equations of the characteristic curves, and hence show that the characteristic coordinates may be chosen as (= x² - y², þ= y. = 0. - = 0 (x > 0). (c) Show that the corresponding standard form for the equation is J²u მმი - (d) By solving the standard form given in part (c), find the general solution of the original equation.
The function u(x, y) satisfies the partial differential equation J²u J²u xy- əx² дхду +x². ди Y əx (a) Show that this equation is hyperbolic. (b) Find the equations of the characteristic curves, and hence show that the characteristic coordinates may be chosen as (= x² - y², þ= y. = 0. - = 0 (x > 0). (c) Show that the corresponding standard form for the equation is J²u მმი - (d) By solving the standard form given in part (c), find the general solution of the original equation.
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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Transcribed Image Text:The function u(x, y) satisfies the partial differential equation
J²u
J²u
xy əx²
əxəy
+x².
ди
əx
=
= 0.
0 (x > 0).
(a) Show that this equation is hyperbolic.
(b) Find the equations of the characteristic curves, and hence show
that the characteristic coordinates may be chosen as
<= x² - y², 0 = y.
(c) Show that the corresponding standard form for the equation is
J²u
მმი
(d) By solving the standard form given in part (c), find the general
solution of the original equation.
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