Final answer:
The type of the PDE yuxx - xuxy + yuyy = 0 is determined by the discriminant D = x2/4 - 4y2 and can be elliptic, hyperbolic, or parabolic depending on the values of x and y.
Step-by-step explanation:
The given second-order linear partial differential equation (PDE) is yuxx - xuxy + yuyy = 0. To determine the type of this PDE, we look at the coefficients of the second derivative terms and form the discriminant D = b2 - 4ac from the general second-order linear PDE Auxx + 2Buxy + Cuyy + ... = 0, where A, B, and C are constants or functions of x and y. Here, A = y, B = -x/2, and C = y. The discriminant is thus D = (-x/2)2 - 4(y)(y) = x2/4 - 4y2. Depending on the sign of D, the PDE can be classified as elliptic (D < 0), hyperbolic (D > 0), or parabolic (D = 0). In this case, the sign of D depends on the values of x and y, and so the PDE may change type in different regions of the xy-plane.