Explanation:
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the
function itself and its derivatives of various orders. Differential equations play a prominent role in
engineering, physics, economics, and other disciplines.
An example of
modelling a real world problem using differential equations is the
determination of the velocity of a ball falling through the air, considering
only gravity and air resistance. The ball acceleration towards the ground is
the acceleration due to gravity minus the deceleration due to air resistance.
Gravity is considered constant, and air resistance may be modelled as
proportional to the ball's velocity. This means that the ball's acceleration,
which is the derivative of its velocity, depends on the velocity. Finding the
velocity as a function of time involves solving a differential equation.
Differential
equations are mathematically studied from several different perspectives, mostly
concerned with their solutions -the set of functions that satisfy the equation.
Only the simplest differential equations admit solutions given by explicit
formulas; however, some properties of solutions of a given differential
equation may be determined without finding their exact form. If a
self-contained formula for the solution is not available, the solution may be
numerically approximated using computers.
http://en.wikipedia.org/wiki/Differential_equation
In mathematics, an integrating factor is a function that is chosen to facilitate
the solving of a given equation involving differentials. It is commonly used to
solve ordinary differential
equations, but is also used within multivariable calculus, in this
case often multiplying through by an integrating factor allows an inexact differential to be made into an exact differential(which can
then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the
integrating factor that makes entropy an exact differential.
Use in
solving first order linear ordinary differential equations
Consider an ordinary differential
equation of the form
Consider a function M(x). We
multiply both sides of (1) by M(x):
We want the
left hand side to be in the form of the derivative of a product (see product rule), such that (2)
can be written as
The left hand side in (3) can now be integrated
We can now solve for y,
Applying the product rule to the left hand side of (3)
and equating to the left hand side of (2)
From which it is clear that M(x) obeys
the differential equation :
Solving (5) gives
M(x) is
called an integrating factor.
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Integrating_factor
The
above explanation is copied from Wikipedia, the free encyclopedia and is
remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported
License.