2nd Order Liner PDE’s Part 1

a) Determine whether the given eqn is hyperbolic, parabolic or elliptic

b) Determine the subset of R^2

c) Determine the characteristics curves

$\displaystyle

u_{xx}+2xu_{yx}+u_{yy}+Cos(xy)u_{y}=u(x,y)

$

$\displaystyle

B^2-AC = x^2-1(1) = x^2-1

$

a)

Hyperbolic if $\displaystyle x^2-1 > 0 $

ie $\displaystyle x^2>1 $

$\displaystyle

x>1, x<-1

$

Elliptic if $\displaystyle x^2-1< 0$

ie $\displaystyle x^2< 1 $

$\displaystyle

-1<x<1

$

Parabolic if $\displaystyle x^2-1 = 0$

ie $\displaystyle x^2=1 $

$\displaystyle

x=-1, x=+1

$

b) How do you write the subset query correctly?

c)

$\displaystyle

\frac{dy}{dx} = \frac{B \pm \sqrt{B^2-AC}}{A} = x \pm \sqrt{x^2-1}

$

My initial attempt to solve for y on this by letting u = x^2 -1 and use substitution doesnt seem to work?

Thanks