Answer to Question #199540 in Differential Equations for Rajkumar

Question

Find the value of n for which the equation (n - 1)² u_xx - y²ⁿ u_yy = ny^2n-1 u_y is parabolic or hyperbolic.

Answer 1

Find the value of $n$ for which the equation $(n-1)^2u_{xx}-y^{2n}u_{yy}=ny^{2n-1}u_y$ is

(1) parabolic

(2) hyperbolic

Second-order PDE

$$A(x,y)u_{xx}+B(x,y)u_{xy}+C(x,y)u_{yy}=F(x,y,u,u_x,u_y)$$

The type of second-order PDE at a point $(x_0,y_0)$ depends on the sign of the discriminant defined as

$$\Delta(x_0,y_0)=\begin{vmatrix} B & 2A \\ 2C & B \end{vmatrix}$$ $$\Delta(x_0,y_0)=\begin{vmatrix} 0 & 2(n-1)^2 \\ 2(-y^{2n}) & 0 \end{vmatrix}=4(n-1)^2y^{2n}$$

(1) parabolic: $\Delta(x_0,y_0)=0=>4(n-1)^2y^{2n}=0=>n=1 \ or\ y=0$

$n=1 \ or\ y=0$

(2) hyperbolic: $\Delta(x_0,y_0)>0=>4(n-1)^2y^{2n}>0=>n\not=1, y\not=0$

$n\not=1, y\not=0$