Question

Consider the second order linear partial differential equation a(x, y)uxx + 2b(x, y)uxy + c(x, y)uyy + d(x, y)ux + e(x, y)uy + f(x, y)u = 0. (1)

a). Show that under a coordinate transformation ξ = ξ(x, y), η = η(x, y) the PDE (1) transforms into the PDE α(ξ, η)uξξ + 2β(ξ, η)uξη + γ(ξ, η)uηη + δ(ξ, η)uξ + (ξ, η)uη + ψ(ξ, η)u = 0, (2) where α(ξ, η) = aξ2 x + 2bξxξy + cξ2 y , β(ξ, η) = aξxηx + b(ξxηy + ηxξy) + cξyηy, γ(ξ, η) = aη2 x + 2bηxηy + cη2 y , δ(ξ, η) = aξxx + 2bξxy + cξyy + dξx + eξy, (ξ, η) = aηxx + 2bηxy + cηyy + dηx + eηy, ψ(ξ, η) = f

b). Show that the coordinate transformation ξ(x, y) = y/x and η = η(x, y) arbitrary, brings the PDE x 2uxx + 2xyuxy + y 2uyy + xyux + y 2uy = 0 (3) into the form (2) with α = β = δ = ψ = 0.

c). Find the general solution of (3) by a judicious choice of η = η(x, y) in b).

Answer 1