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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
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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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: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.
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
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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