Question: uestion 1: Consider the second order linear partial differential equation a(x, y)uxx + 2b(x, y)ury + c(x, y)Uyy + d(x, y)ur te(x, y)Uy + f(x,

uestion 1: Consider the second order linear partial differential equation a(x, y)uxx + 2b(x, y)ury + c(x, y)Uyy + d(x, y)ur te(x, y)Uy + f(x, y)u = 0. (1) a). Show that under a coordinate transformation $ = (x, y), n = n(x, y) the PDE (1) transforms into the PDE a (5, n )usE + 2 B( 5, n)uEn ty (5, n)unn + 6(5, n)us + E(5, n)un + 2(5, n)u= 0, (2) where a(5, n) = aEx + 2bExEy + c$2 B(5, n) = aExx + b(Exny + nxy) + cSynly, 7 (5, n ) = anx + 2bnxy + cny, 8(5, n) = aExx + 2bExy + CEyy + dEx tely, E(S, n) = anxx + 2bnry + Cnyy + anx + eny, 2/ ( 5, n) = f

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