Editing Hyperbolic partial differential equation (section)

== Examples ==

By a linear change of variables, any equation of the form
<math display="block"> A\frac{\partial^2 u}{\partial x^2} + 2B\frac{\partial^2 u}{\partial x\partial y} + C\frac{\partial^2u}{\partial y^2} + \text{(lower order derivative terms)} = 0</math>
with
<math display="block"> B^2 - A C > 0</math>
can be transformed to the [[wave equation]], apart from lower order terms which are inessential for the qualitative understanding of the equation.<ref name="Evans 1998"/>{{rp|p=400}}  This definition is analogous to the definition of a planar [[Hyperbola#Quadratic equation|hyperbola]].

The one-dimensional [[wave equation]]:
<math display="block">\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0</math>
is an example of a hyperbolic equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE. This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.<ref name="Evans 1998">{{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | orig-year=1998 | url=https://www.worldcat.org/oclc/465190110 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=[[Graduate Studies in Mathematics]] | isbn=978-0-8218-4974-3 |mr=2597943 | year=2010 | volume=19 | doi=10.1090/gsm/019| oclc=465190110 }}</ref>{{rp|p=402}}