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diffrential equations

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agtsp

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I have fundamental doubt on differential equations. Let us say that we want to solve dy/dx+2y=0 then by traditional method we get the solution e power (-2x). Now the question is two fold, which come first, the solution or the differential equation. I mean that in practice we would take measurements and plot a graph and that would represent a function. Now how do I frame a differential equation from a function?
(example how can I say by looking at e power(-2x) that dy/dx+2y=0 will be the governing equation)
 

As far as I understood your question, you know that y=e(-2x) then dy/dx=-2*e(-2x). Now just substitute in your differential equation:

-2*e(-2x) + 2*e(-2x) = 0
 

Simple terms speaking just differentiating and adding terms of y that will turn the whole equation to 0

If sine and cos are involved we go for double differentiation and thus a second order DE is arrived

But there is no fixed rule that a sine function has to be in second order and all
 

If you are given the differential equation, you can find the solution - the equation in x,y form. I have shown that as sum 1. (By the way, you forgot to take into consideration the constant of integration).
If you are given the equation in x,y form, you can find the differential equation. I have shown that as sum 2.

3500937600_1352563836.png


Hope this helps.
Tahmid.
 
Actually let me reframe my question. How do we model real life situation in terms of differential equation. Let me an example. Take newtons law of cooling. If we just note down the readings and plot it, we get an exponential curve and that is the solution we are looking for. Certainly we do not need a differential equation to describe the newtons law of cooling because we already have a solution. In that case when do we need a differential equation at all?
 

When you plot the readings you have a description of what is recorded.. Unfortunately, you wont be able to predict the results for the same phenomena in different conditions for example a more complex geometry, a different material, or combination of materials, and that is why Differential equations comes in handy. Imagine that you can Analise what is happening in a small fraction of a system, that you can model a mathematic law for that infinitesimal part of the system... that model will be a differential equation... Now you just need to adjust that model to the rest of the system, either by integration or differential calculus, meaning you apply that same law to the rest of system..
 

As a simple example: a capacitor's current and voltage relation is a differential relation, i=c*dv/dt, why it is differential... you study. In a simple RC circuit, if you write Kirchhoff laws, you end up with a differential equation. There are many phenomena having differential relationship in natura. Differential relations bring differential equations.
 

agtsp said:
Actually let me reframe my question.
...
In that case when do we need a differential equation at all?
Click to expand...

The tendency is to use the easy formula. Often it turns out to be sufficient.

Example, when we think of non-linear components, the PN junction comes to mind. Diodes, zener diodes, and LED's. To calculate their response, it just so happens that their behavior is more or less centered around the threshold voltage. The diode equation (which contains an exponent or two) rarely needs to be brought into the picture.

The transistor base has a PN junction. Once its threshold is reached, the response parameters can be calculated in algebra.

All this is to say, we can often get away with approximating a performance curve as a straight line.
 

Hi mgate

Yes, I agree with your answer. But If you could you give me an example, it will aid in my clear understanding. Taking the very same example of Newtons law of cooling, with the solution you can predict for any situation. Or for that matter, a simple RC circuit, the solution is so very generalized that , the prediction of the output is easy for any value for R and C. Hence I do not see any reason for representing a phenomena through differential equations unless the parameter is hidden (such as velocity or acceleration) in a function.
 

BradtheRad , That is discrete Differential Equation case.
agtsp , Show me the newton law of Cooling .

And depend on problem modeling for v and a, use PDE .
 

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