Discussion
The equation
\[ 3\Delta u + 4u_{xy} - u^{2} = 1 \]
is :
Answer: D
The equation is elliptic.
Expand \(3\Delta u = 3u_{xx} + 3u_{yy}\), so the principal part is \(3u_{xx} + 4u_{xy} + 3u_{yy}\). For \(Au_{xx} + 2Bu_{xy} + Cu_{yy}\), the type depends on \(B^{2} - AC\):
\[ B^{2} - AC = 2^{2} - 3 \cdot 3 = -5 \lt 0 \]
A negative discriminant means elliptic. The term \(-u^{2}\) makes the equation non-linear (so 'Linear' is wrong), but it is a lower-order term and does not change the type. Hyperbolic would need a positive and parabolic a zero discriminant.
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