Discussion
The partial differential equation :
\[ x^{2}u_{xx} - 2xyu_{xy} + y^{2}u_{yy} + xu_{x} + yu_{y} = 0 \]
is :
Answer: D
The equation is parabolic on \(\mathbf{R}^{2}\).
For \(Au_{xx} + 2Bu_{xy} + Cu_{yy} + \ldots = 0\) the type is decided by the sign of \(B^{2} - AC\). Here \(A = x^{2}\), \(2B = -2xy\) so \(B = -xy\), and \(C = y^{2}\):
\[ B^{2} - AC = x^{2}y^{2} - x^{2}y^{2} = 0 \]
The discriminant vanishes at every point, so the equation is parabolic throughout the plane. (Indeed the principal part is \((x\partial_{x} - y\partial_{y})^{2}\) up to lower-order terms, a perfect square.)
Hyperbolic would need \(B^{2} - AC \gt 0\) and elliptic \(B^{2} - AC \lt 0\); neither happens anywhere, including the first quadrant singled out in the first option.
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