# differential equation

A differential equation is a description of how something continuously changes over time. Differential equations occur in many, if not most, physical
problems. Some differential equations have an exact **analytical solution** such that all future states can be known without simulating the time evolution
of the system. However, most have a **numerical solution** with only limited accuracy.

A differential equation involves the first or higher derivatives of the function to be solved for. If the equation
only involves first derivatives it is known as an equation of *order* one, and so on. If only *n*-th powers of the derivatives are involved,
the equation is said to have *degree n*. Equations of degree one are
called *linear*. Equations in only one variable are called ordinary
differential equations to distinguish them from partial
differential equations, which have two or more.

## Example of a differential equation and its solution

Consider an object accelerating (see acceleration)
uniformly at 40 m/s^{2}. After a time *t* it has a velocity 40*t*, assuming a stationary start. This velocity may be expressed
as *ds*/*dt*, the instantaneous rate of change of distance, *s*. Thus

40*t* = *ds*/*dt*

To find out the distance traveled by the object after a time *t* we can integrate to find

20*t*^{ 2} + *k* = *s* + *c*

or *s* = 20*t*^{ 2} + (*k* - *c*)

where *k* and *c* are constants. However, we have assumed
a stationary start, i.e., that when *t* = 0, *s* = 0 and hence,
by substitution, (*k* - *c*) = 0. Therefore, to find out how
far the body has traveled after a given period of time, we need merely to
substitute the value of *t* into

*s* = 20*t *^{2}

This is the solution of a very simple first-order differential equation.

## Higher order differential equations

In some problems there occur second derivatives of the form *d*^{ 2}*y*/*d**x*^{2}, and these involve solutions
of **second-order differential equations**. Equations involving *n*th derivatives, *d ^{ n}y*/

*dx*, are called

^{n}*n*th-order equations, most important of which are equations of the form

where A, B, C, ..., N are constants. This is termed a **linear nth-order
differential equation**.