4. What is the type of the equation Uxx - 4U xy +4Uyy = 0? Show by direct substitution that u(x, y) = f(y + 2x) + xg(y + 2x) is a solution for arbitrary functions f and g.
4. What is the type of the equation Uxx - 4U xy +4Uyy = 0? Show by direct substitution that u(x, y) = f(y + 2x) + xg(y + 2x) is a solution for arbitrary functions f and g.
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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Question
[Second Order Equations] How do you solve this?

Transcribed Image Text:4. What is the type of the equation
Uxx − 4Uxy +4Uyy = 0?
Show by direct substitution that u(x, y) = f(y + 2x) + xg(y + 2x) is a
solution for arbitrary functions ƒ and g.

Transcribed Image Text:Theorem 1. By a linear transformation of the independent variables, the
equation can be reduced to one of three forms, as follows.
(1)
Elliptic case: If a12 < a11922, it is reducible to
Uxx + Uyy +... 0
(where... denotes terms of order 1 or 0).
(ii) Hyperbolic case: If a²2 > a11922, it is reducible to
UxxUyy +
= 0.
1.6 TYPES OF SECOND-ORDER EQUATIONS
(iii) Parabolic case: If a 12:
= a11922, it is reducible to
Uxx + = 0
(unless a11 = a12 = a22 = 0).
29
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