91Ó°ÊÓ

Whether a second-order linear partial differential equation in \(u=u(x, y)\) is hyperbolic, elliptic, or parabolic (Sec. 7) can vary from region to region in the \(x y\) plane when at least one of the coefficients is a nonconstant function of \(x\) and \(y\). Classify each of the following differential equations in various regions, and sketch those regions. (a) \(y u_{x x}+u_{y y}=0 ; \quad\) (b) \(u_{x x}+2 x^{2} u_{x y}+y u_{y y}=0 ;\) (c) \(x u_{x x}+y u_{y y}-3 u_{y}=2 ; \quad(d) u_{x x}-2 x u_{x y}+\left(1-y^{2}\right) u_{y y}=0\). Answers: \((a)\) Parabolic on the \(x\) axis, elliptic above it, and hyperbolic below it; (b) parabolic on the curve \(y=x^{4}\), elliptic above it, and hyperbolic below it; (d) parabolic on the circle \(x^{2}+y^{2}=1\), elliptic inside it, and hyperbolic outside it.