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 http://mathforum.org/dr.math/ 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. |
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