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Old Jul 28th 2008, 10:03 AM   #1
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Differential equations

I found that there are many applications of differential equations in physics. Cound anyone please tell me what is the importance of these equations and what things could we benefit from that?
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Old Jul 28th 2008, 12:57 PM   #2
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Originally Posted by werehk View Post
I found that there are many applications of differential equations in physics. Cound anyone please tell me what is the importance of these equations and what things could we benefit from that?
You have asked an interesting question. Newton invented and applied Calculus to describe the motion of a mass due to gravity. (Don't say that in front of a Mathematician!) Except for the discovery and utility of Calculus this is not such an amazing thing in Physics; since that time practically all applications of Physics have used differential and integral Calculus.

Calculus is based on two operations: taking the derivative of a function and integrating a function. They are (mostly) inverse operations of each other. The derivative is nothing more than a "instantaneous rate of change" which comes in very handy as you can see. (Given the questions you are asking I'm presuming you've taken at least two semesters of Physics anyway.) The differential equations come in when we are considering mulitple derivatives of a function and how they relate to one another. Most of Physics turns out to be fairly simple in my opinion. Applying that Physics and translating it into a Mathematical expression also tends to be fairly simple unless you are right on the "edge" of current research. It's when you try to solve that equation that you start running into problems. For example when writing out the equation for the motion of a falling object with a drag force proportional to the speed of the object, the Physics work is nothing more complicated than a simple application of Netwon's 2nd Law to write the equation. Solving the equation for the first time often ties students' patience into knots.

As far as benefits are concerned differential equations often makes the translation of the Physics to the Mathematics simpler. Many Physics problems are easier to consider on an "instant by instant" basis. So if we can write the problem in these terms, which often involve derivatives, we can often immediately write down the differential equation that the process "obeys" without any further strenuous work, even if the final equation representing the process is hideously complicated.

I say that this question is interesting because I have noted the following fact: the Universe does not solve differential equations. It does what it does. Our models of how the Universe works depends on differential equations to predict the final results but the Universe works what it does and does not have to go through that translation process. Just what the Universe does is still a mystery that is often ignored for the sake of being able to make the predictions.

-Dan
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