## differential equationA 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 solutionConsider an object accelerating (see acceleration) uniformly at 40 m/s^{2}. After a time t it has a velocity
40t, assuming a stationary start. This velocity may be expressed
as ds/dt, the instantaneous rate of change of distance,
s. Thus
40To find out the distance traveled by the object after a time t
we can integrate to find
20where 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
This is the solution of a very simple first-order differential equation. ## Higher order differential equationsIn some problems there occur second derivatives of the formd^{
2}y/dx^{2}, and these involve solutions
of second-order differential equations. Equations involving
nth derivatives, d/^{ n}ydx,
are called ^{n}nth-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## Related entry• difference equation## Related category• CALCULUS AND ANALYSIS | |||||

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