Discussion
The partial differential equation
\[ \begin{aligned} &\frac{\partial^{2}u}{\partial x^{2}} + 4\frac{\partial^{2}u}{\partial x \partial y} \\ &+ 2\frac{\partial^{2}u}{\partial y^{2}} = 0 \end{aligned} \]
is ______.
Answer: B
The equation is hyperbolic.
A second-order equation \(Au_{xx} + 2Bu_{xy} + Cu_{yy} + \dots = 0\) is classified by \(B^{2} - AC\) (equivalently by \(b^{2} - 4ac\) when the mixed coefficient is written as b). Here the coefficients are 1, 4 and 2:
\[ b^{2} - 4ac = 4^{2} - 4(1)(2) = 8 \gt 0 \]
A positive discriminant means hyperbolic, zero means parabolic and negative means elliptic. The coefficients are constants, so the type is the same everywhere and the equation is not of mixed type.
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