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Example:Solving Homogeneous Differential Equation

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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.




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)




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)




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.