Question

Find the value of `n` for which the equation `(n-1)^(2)u_{x x}-y^(2n)u _{yy}=ny^(2n-1)u_y` is parabolic or hyperbolic?

Answer 1

See below.

Explanation:

Given a second order PDE

# A(x,y)u_{x x}+B(x,y) u_{xy}+C(x,y)u_{yy} = \Phi(x,y,u,u_x,u_y) #

according to the sign of

# \Delta = \det((B,2A),(2C,B)) = B^2-4AC #

we qualify the PDE hence

# \Delta = 4(n-1)^2y^{2n} #

which for `(n \ne 1) \cap (y \ne 0)\Rightarrow \Delta > 0`

In those conditions the PDE is of hyperbolic kind otherwise is parabolic.