Separation of Variables?

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apple2357

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I know there is a standard method to solve differential equations using separation of variables. I am trying to understand why this method is valid? I know that technically you cannot separate out dy and dx and rearrange but it seems to work. Why does it work?
My own thoughts on this is that it appears to combine the chain rule with perhaps implicit differentiation in some senses but i can't quite pin it down.
Any help please?
 
apple2357 said:
I know there is a standard method to solve differential equations using separation of variables. I am trying to understand why this method is valid? I know that technically you cannot separate out dy and dx and rearrange but it seems to work. Why does it work?
My own thoughts on this is that it appears to combine the chain rule with perhaps implicit differentiation in some senses but i can't quite pin it down.
Any help please?
Click to expand...
Perhaps an example?
There are a few things you may encounter on the way to such a solution. 1) Algebra, 2) Substitution -- These are popular.
 
I meant the general approach to solve a differential equation: dy/dx= f(x)g(y) which appears to involve treating dy/dx as a fraction! And separating out the variables.
 
Well, sure. It the variables are already separated, you're almost done. Those differentials are pretty versatile.

dy/dx -- A symbol that mean a derivative of a certain function with respect to a certain variable.
dy/g(y) = f(x) dx -- Differential indicators indicating the variable of interest for integration.
dy = h'(x) dx -- Differentials indicating some small value for linear approximation.

It can be a shock to some students when the notation breaks out of where you want it to stay. Just be clear.
 
apple2357 said:
I meant the general approach to solve a differential equation: dy/dx= f(x)g(y) which appears to involve treating dy/dx as a fraction! And separating out the variables.
Click to expand...

This explanation of solving separable differential equations shows the mathematical justification for the process: http://tutorial.math.lamar.edu/Classes/DE/Separable.aspx .

It is one of several cases where, although dy/dx is not really a fraction, it can be proved that it is valid (in these specific situations) to treat it as if it were. That is, in moving the dx and dy around, you are actually carrying out a shortcut for a provable theorem. But it is so convenient that we generally just do it without worrying about the justification.
 
This is an excellent question. Good for you to think about it! We also just do it when we make u-substitutions in calc 2. It has been years since I thought about this. Dr Peterson is so correct when he wrote that we generally just do it without worrying about the justification
 
I can add that once we know it is justified, it's fine to just use the rule -- just as we can apply any theorem after it has been proved. Unfortunately, it seems that a good number of teachers just state the rule without mentioning that it can be justified by something more valid than just the fact that it feels right to move dx and dy around like that.

As Jomo said, you're wise to have asked. You have the makings of a mathematician.
 
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