Math Problem Statement
Obtain the general solution of the following equation x²uxx+2xyuxy+y²uyy+xyux+y²uy=0
Solution
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Math Problem Analysis
Mathematical Concepts
Partial Differential Equations
Euler-Cauchy Equations
Separation of Variables
Formulas
General form of a second-order partial differential equation: a(x, y)u_{xx} + 2b(x, y)u_{xy} + c(x, y)u_{yy} + d(x, y)u_{x} + e(x, y)u_{y} + f(x, y)u = 0
Separable solution assumption: u(x, y) = x^m y^n
Characteristic equation: (m + 1/2)^2 + (n + 1/2)^2 = 1/2
Theorems
Method of Separation of Variables
Euler-Cauchy Homogeneous Differential Equations
Suitable Grade Level
University Level (Advanced Calculus/Mathematics)
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