formula for the checking of the PDE to be hyperbolic, elliptic, parabolic?

$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$ the formula that my book uses to check for the conditions for the elliptic, hyperbolic and the parabolic equation is $$b^2-4ac\gt 0$$ for hyperbolic equations, $$b^2-4ac\lt 0$$ for elliptic equations and $$b^2-4ac=0$$ for the parabolic equation.
But on wiki it is mentioned to be by using $$b^2-ac,$$ is the result given in my book is wrong. What is the soul derivation of saying about these equations to be hyperbolic, parabolic, elliptic?

• Maybe that in your book the initial formula have a $B$ and not a $2B$ at the second term? Apr 24, 2016 at 14:25
• suppose i have a equation $u_{xx} + 2u_{xy} + u_{yy} = 0$ then to to find out which type of equation it is i can use $2^2-4=0$ @emilio novati Apr 24, 2016 at 17:07

The definition in Wikipedia is correct and it is reffer to an equation written in the form: $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ , but I suspect that also the definition in you book is correct and it is referred to an equation written in the form: $$Au_{xx} + B'u_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$
Note that, in this case, we have: $$(B')^2-4AC=(2B)^2-4AC=4(B^2-AC)$$