TITLE: Why are certain PDE called "elliptic", "hyperbolic", or "parabolic"?
QUESTION [38 upvotes]: Why are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these, but I don't understand why they are so named?
Does it has anything to do with the ellipse, hyperbolas and parabolas?

REPLY [22 votes]: I'd like to address this comment of the OP to one of the answers:

But I think the solutions of the corresponding differential equations have nothing to do with their name? Or do they have?.

Actually, we can relate the names to the corresponding geometrical curves. To do this, consider how the following homogeneous PDE will look in the Fourier-transformed form.
Original PDE (with $u^{(n,m)}(x,y)$ denoting $n$th partial derivative of $u$ in $x$ and $m$th in $y$):
$$Au^{(2,0)}(x,y) + 2Bu^{(1,1)}(x,y) + Cu^{(0,2)}(x,y) + Du^{(1,0)}(x,y) + Eu^{(0,1)}(x,y) = 0.\tag1$$
Fourier-transformed one (with $\hat u(k_x,k_y)$ denoting the Fourier transform of $u(x,y)$):
$$\mathcal L\hat u(k_x,k_y) = 0,\tag2$$
where
$$\mathcal L=Ak_x^2 + 2Bk_xk_y + Ck_y^2 + Dk_x + Ek_y.\tag3$$
This multiplication by $\mathcal L$ is the Fourier-space version of the differential operator from $(1)$. Notice that $\mathcal L$ is just a second-degree polynomial in $k_x$, $k_y$ — a quadratic form. Now consider the nodes of $\mathcal L$ in the $(k_x,k_y)$ plane: they will be defined by the equation
$\mathcal L=0.\tag4$
Geometrically, these nodes will be exactly the kinds of curves with the names corresponding to the name of the kind of PDE, i.e. ellipses, hyperbolas or parabolas.