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
Homogeneous differential
equation
The term homogeneous
differential equation has
several distinct meanings.
One meaning is that a first-order ordinary differential equation is homogeneous (of degree 0) if it has
the form
where F(x,y) is a homogeneous
function of degree zero; that is
to say, such that F(tx,ty)
= F(x,y).
In a related, but distinct, usage, the term linear homogeneous differential
equation is used to describe differential equations of the form
where the differential
operator L is a linear
operator, and y is the unknown function.
The remainder of this article is about homogeneous differential
equations in the first sense defined above.
Solving
homogeneous differential equations
By the
definition above, it can be seen that F(tx,ty) = F(x,y) for all t, so t can be arbitrarily chosen to
simplify the form of the equation. One can solve this equation by making a
simple change of variables y = ux, and then using the product rule on the left hand side as
follows,
and then using the identity F(tx,ty)
= F(x,y) to simplify the right hand side by choosing to set t to be 1
/ x, transforming the
original problem into the separable differential equation
which can then be integrated by the usual methods.
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Homogeneous_differential_equation
Separation of variables
In mathematics, separation of variables is any of several methods for
solving ordinary and partial differential equations, in which algebra allows one to rewrite an
equation so that each of two variables occurs on a different side of the
equation.
Ordinary
differential equations (ODE)
Suppose a differential equation can be
written in the form
which we can write more simply by
letting y = f(x):
As long as h(y)
≠ 0, we can rearrange terms to obtain:
so that the two variables x and y have
been separated. dx (and dy) can be viewed, at a
simple level, as just a convenient notation, which provides a handy mnemonic
aid for assisting with manipulations.
A formal definition of dx as a differential (infinitesimal) is
somewhat advanced.
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Separation_of_variables
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above explanation is copied from Wikipedia, the free encyclopedia and is
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License.