Question

find the value of n for which the equations (n-1)^2 Uxx - y^2n Uyy= ny^2n-1 is parabola and hyperbola

Answer 1

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

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

$\Delta(x_o,y_o)=\begin{vmatrix} B & 2A \\ 2C & B \end{vmatrix}$

$\Delta(x_o,y_o)=\begin{vmatrix} 0 & 2(n-1)^2 \\ 2(-y^{2n})& 0 \end{vmatrix}=4(n-1)^2y^{2n}$

For Parabola;

$4(n-1)^2y^{2n}=0$

$n=1$ or $y=0$

For Hyperbola;

$\Delta(x_o,y_o)>0$

$4(n-1)^2y^{2n}>0$

$n\neq1,y\neq0$