$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$
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?
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3$\begingroup$ Maybe that in your book the initial formula have a $B$ and not a $2B$ at the second term? $\endgroup$ – Emilio Novati Apr 24 '16 at 14:25
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$\begingroup$ 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 $\endgroup$ – Bogorovich Apr 24 '16 at 17:07
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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: $$ (B')^2-4AC=(2B)^2-4AC=4(B^2-AC) $$
answered Apr 24 '16 at 17:27
