Mathguru – Help – Find the solution of the given linear differential equation.

9.6 Linear Differential Equations > No. 6

Find the solution of the given linear differential equation.

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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

Integrating factor

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

$Description: y'+P(x)y = Q(x)\quad\quad\quad (1)$

Consider a function M(x). We multiply both sides of (1) by M(x):

$Description: M(x)y' + M(x)P(x)y = M(x)Q(x).\quad\quad\quad (2)$

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

$Description: (M(x)y)' = M(x)Q(x).\quad\quad\quad (3)$

The left hand side in (3) can now be integrated

$Description: y M(x) = \int Q(x) M(x)\,dx,$

We can now solve for y,

$Description: y = \frac{\int Q(x) M(x)\, dx}{M(x)}.\,$

Applying the product rule to the left hand side of (3) and equating to the left hand side of (2)

$Description: M'(x) y + M(x) y' = M(x) y' + M(x) P(x) y .\quad\quad\quad$

From which it is clear that M(x) obeys the differential equation :

$Description: M'(x) = M(x)P(x) .\quad\quad\quad (4)\,$

$Description: \frac{M'(x)}{M(x)} = P(x) .\quad\quad\quad (5)$

Solving (5) gives

$Description: M(x)=e^{\int P(x)\,dx}.$

M(x) is called an integrating factor.

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Integrating_factor

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