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An Ordinary Differential Equation in Everyday Life?

Date: 03/26/2011 at 05:33:53
From: Hassan
Subject: ODE's

I just want to ask about ordinary differential equations. Where do these
find practical applications?

Date: 03/26/2011 at 20:45:22
From: Doctor Jordan
Subject: Re: ODE's

Hi Hassan,

Let's say I have a small cube of metal of mass m sitting on a table. Say
the cube of metal is attached to a spring in such a way that the cube will
move back and forth on the table.

Suppose we want to know the position of this cube of metal at time t. How
can we obtain this?

This is the kind of problem that call for ordinary differential equations.
Methods for solving ODE let us write down an equation using data
that we have, and then the solution to the ODE will give us the
information that we want -- in this case, the position of the cube at
time t.

To get the differential equation for this mechanical system, we need to
know the forces acting on the object at any time -- namely, the restoring
force of the spring, which is approximately -kx, where x is the distance
from the rest position of the cube and k is an experimentally determined
number that depends on the spring's material composition and
manufacturing; and the damping force, which is approximately cx', where c
is called the viscous damping coefficient. Here, x' is the derivative of x
with respect to t.

Then by Newton's second law here, the force cx' - kx is equal to mx'',
where x'' is the second derivative of x with respect to t -- in other
words, the acceleration of the metal cube. So here the differential
equation is 

   mx'' = cx' - kx
It is called a differential equation because it involves derivatives. It
says how the system is moving at each point in time.

We have mathematical techniques for solving ordinary differential
equations. We apply these techniques to this equation and we get a
solution, x(t). But what is x(t)? x(t) is the position of the metal cube
at time t. Therefore, solving this differential equation gives us a
function which tells us where the metal cube is at any point in time.

We started knowing only the forces acting on the cube at each point in
time and put this together into a differential equation. By applying
mathematical techniques, we ended up with a function that told us the
exact arrangement of our system at each point in time.

- Doctor Jordan, The Math Forum 

Date: 03/27/2011 at 00:27:38
From: Hassan
Subject: Thank you (ODE's)

Nice job; very grateful.

May Allah bless you more.

Jazak Allah.
Associated Topics:
High School Calculus

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