# 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?

Aug 12, 2018

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 $\left(n \setminus \ne 1\right) \setminus \cap \left(y \setminus \ne 0\right) \setminus R i g h t a r r o w \setminus \Delta > 0$

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