Well these are my personal view on what role differential equations have got to play.... read on if u think its worth...
As humans we are mostly interested in knowing what will happen to some thing in the future... What will happen to him? what will be the weather in 2 days, or some more thing else... the whole point is simple...
"I want to know what is going to happen..."
So what is it that a person gets to start off on prediction?
He has seen what has happened in the past, he can think what are the dependant and the independent parameters based on this experience.... he continues from here and makes an assumption as to the directly varying or indirectly varying and then comes up with an equation....
Well this was a crude that worked wonderfully with the initial understanding....
but, what the person currently sees is only a small portion of the whole, the smallest part of the system almost negligibly small instance that he considers... so at this particular instant if he has to form an equation it is better he forms a differential eqn, that is a better way to write it down.... so then if he has a differential eqn, and wants to know what happens in the long run, he just has to integrate and find out...
An example:
well now this is how I start my prediction... if the change is always a constant... lets say...
we want to measure some distance.... I see that I have been travelling some 3mt in 2 seconds... for a consistent amount of time.... so now I say this is a constant for every small interval of time say a second... I travel 1.5 mt So now I continue to write the differential eqn...
dx/dt = 1.5
now I integrate to form my equation.... X = 1.5t... I continue this way to form the eqn...
well there is a catch - in writing these dif-eqn... most of these are unsolvable... that's coz we cant predict to the minutest detail.... true rt!!!
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