A constant-coefficient second-order partial differential
Chapter 10, Problem 3TWE(choose chapter or problem)
A constant-coefficient second-order partial differential equation of the form
\(a \frac{\partial^{2} u}{\partial x^{2}}+b \frac{\partial^{2} u}{\partial x \partial y}+c \frac{\partial^{2} u}{\partial y^{2}}=0\)
can be classified using the discriminant \(D:=b^{2}-4 a c\). In particular, the equation is called
hyperbolic if \(D>0\), and
elliptic if \(D<0\)
Verify that the wave equation is hyperbolic and Laplace's equation is elliptic. It can be shown that such hyperbolic (elliptic) equations can be transformed by a linear change of variables into the wave (Laplace's) equation. Based on your knowledge of the latter equations, describe which types of problems (initial value, boundary value, etc.) are appropriate for hyperbolic equations and elliptic equations.
Equation Transcription:
Text Transcription:
\partial^2 u\partial x^2+b \partial^2 u\partial x \partial y+c \partial^2 u\partial y^2=0
D:=b2-4ac
D>0
D<0
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