Discussion
The region in which the partial differential equation,
\[ yu_{xx} + 2xyu_{xy} + xu_{yy} = u_{x} + u_{y} \]
is hyperbolic if :
Answer: C
The equation is hyperbolic where \(xy \lt 0\).
For \(A u_{xx} + 2B u_{xy} + C u_{yy} + \ldots = 0\), the type depends on \(B^{2} - AC\). Here \(A = y\), \(2B = 2xy\) so \(B = xy\), and \(C = x\):
\[ B^{2} - AC = x^{2}y^{2} - xy = xy(xy - 1) \]
This is positive (hyperbolic) when \(xy \lt 0\) or \(xy \gt 1\). Of the options, only \(xy \lt 0\) describes a region where the equation is hyperbolic throughout.
\(xy \lt 1\) includes \(0 \lt xy \lt 1\), where \(B^{2} - AC \lt 0\) and the equation is elliptic. \(xy \neq 0\) and \(xy \neq 1\) also include that elliptic band.
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