Let u =u(x, y). (a) Consider the P.D.E. 0= y'u - 2xyu, +x*u, + yu, +...

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Let u =u(x, y). (a) Consider the P.D.E. 0= y'u - 2xyu, +x*u, + yu, + xu,. Identify the equation type (hyperbolic, parabolic or elliptic) [3 marks], and explain [4 marks] the geometric nature of any family/families of characteristics (characteristic curves) associated with the general solution (you are not required to find the general solution): (7 marks) (b) Determine those regions in the x, y plane for which the P.D.E. 8= yu_ - 2u, +e*u,-x'u + 4xu (y20) is hyperbolic, parabolic and elliptic, and indicate them with a sketch [13 marks). Furthermore, in the case when the equation is parabolic, show [5 marks] that the family of characteristics comprises parabolas of general form y =-2x+ 4 (where A is an arbitrary constant). Let u =u(x, y). (a) Consider the P.D.E. 0= y'u - 2xyu, +x*u, + yu, + xu,. Identify the equation type (hyperbolic, parabolic or elliptic) [3 marks], and explain [4 marks] the geometric nature of any family/families of characteristics (characteristic curves) associated with the general solution (you are not required to find the general solution): (7 marks) (b) Determine those regions in the x, y plane for which the P.D.E. 8= yu_ - 2u, +e*u,-x'u + 4xu (y20) is hyperbolic, parabolic and elliptic, and indicate them with a sketch [13 marks). Furthermore, in the case when the equation is parabolic, show [5 marks] that the family of characteristics comprises parabolas of general form y =-2x+ 4 (where A is an arbitrary constant).