[Second Order Equations] How do you solve this? 2. Find the regions in the xy plane where the equation(1+x)Uxx + 2xyuxy - y²uyy = 0is elliptic, hyperbolic, or parabolic. Sketch them. Theorem 1. By a linear transformation of the independent variables, theequation can be reduced to one of three forms, as follows.(1)Elliptic case: If a12 < a11922, it is reducible toUxx + Uyy +... 0(where... denotes terms of order 1 or 0).(ii) Hyperbolic case: If a²2 > a11922, it is reducible toUxxUyy += 0.1.6 TYPES OF SECOND-ORDER EQUATIONS(iii) Parabolic case: If a 12:= a11922, it is reducible toUxx + = 0(unless a11 = a12 = a22 = 0).29

Given: Partial differential equation is 1+xuxx+2xyuxy-y2uyy=0 To Find: Region in XY plane where the…

Answered: 2. Find the regions in the xy plane…

2. Find the regions in the xy plane where the equation (1+x)Uxx +2xyUxy — y²Uyy = 0 is elliptic, hyperbolic, or parabolic. Sketch them.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 18T

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Question

[Second Order Equations] How do you solve this?

2. Find the regions in the xy plane where the equation
(1+x)Uxx + 2xyuxy - y²uyy = 0
is elliptic, hyperbolic, or parabolic. Sketch them.

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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