Editing Differential equation (section)

===Examples===

In the first group of examples ''u'' is an unknown function of ''x'', and ''c'' and ''ω'' are constants that are supposed to be known. Two broad classifications of both ordinary and partial differential equations consist of distinguishing between ''[[linear differential equation|linear]]'' and ''nonlinear'' differential equations, and between [[homogeneous differential equation|''homogeneous'' differential equation]]s and ''heterogeneous'' ones.
* Heterogeneous first-order linear constant coefficient ordinary differential equation:
*: <math> \frac{du}{dx} = cu+x^2. </math>
* Homogeneous second-order linear ordinary differential equation:
*: <math> \frac{d^2u}{dx^2} - x\frac{du}{dx} + u = 0. </math>
* Homogeneous second-order linear constant coefficient ordinary differential equation describing the [[harmonic oscillator]]:
*: <math> \frac{d^2u}{dx^2} + \omega^2u = 0. </math>
* Heterogeneous first-order nonlinear ordinary differential equation:
*: <math> \frac{du}{dx} = u^2 + 4. </math>
* Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a [[pendulum]] of length ''L'':
*: <math> L\frac{d^2u}{dx^2} + g\sin u = 0. </math>

In the next group of examples, the unknown function ''u'' depends on two variables ''x'' and ''t'' or ''x'' and ''y''.
* Homogeneous first-order linear partial differential equation:
*: <math> \frac{\partial u}{\partial t} + t\frac{\partial u}{\partial x} = 0. </math>
* Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the [[Laplace equation]]:
*: <math> \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0. </math>
* Homogeneous third-order non-linear partial differential equation, the [[Korteweg–De Vries equation|KdV equation]]:
*: <math> \frac{\partial u}{\partial t} = 6u\frac{\partial u}{\partial x} - \frac{\partial^3 u}{\partial x^3}. </math>