Discussion
The equation
\[ xu_{xx} - yu_{yy} + \frac{1}{2}(u_{x} - u_{y}) = 0 \]
is hyperbolic in the maximal domain :
Answer: C
The equation is hyperbolic in \(\{(x, y) : xy \gt 0\}\).
For \(Au_{xx} + Bu_{xy} + Cu_{yy} + \ldots = 0\), the type is set by \(B^{2} - 4AC\). Here \(A = x\), \(B = 0\), \(C = -y\):
\[ B^{2} - 4AC = 4xy \]
The equation is hyperbolic where \(4xy \gt 0\), elliptic where \(xy \lt 0\) and parabolic on the axes. The maximal hyperbolic domain is the whole region \(xy \gt 0\), the first and third quadrants together.
Options A and D give only one of the two quadrants, so they are not maximal, and option B, \(x \lt 0, y \gt 0\), is where the equation is elliptic.
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