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# 2011 Calculus AB free response #5c

Solving a differential equation using separation of variables. Created by Sal Khan.

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• Whats the diference between a diferential equation and a implicit derivative. (?)
• Great question!

I'll give you my informal understanding.
They are basically opposites or going in opposite directions.

That is with implicit differentiation you take an ordinary equation like:
x^2 + y^2 = 1
and differentiate it to
2x + 2y dy/dx = 0
and solve for dy/dx.
So we are solving for a derivative.

But the next to last step:
2x + 2y dy/dx = 0
is a differential equation.
And we can take that and solve it and get
x^2 + y^2 = C
(Which is almost what we started with)
With a differential equation we are solving for y or in this case just an ordinary equation with no derivatives.
• i dont understand the theory behind the SEPERATION OF VARIABLES method for integration using dy/dx = g(y).f(x)
1/g(y).dy/dx = f(x)
then integrating both sides ie S 1/g(y).dy/dx .dx =S f(x) .dx w.r.t x
reduces to S 1/g(y).ds = S f(x) .dx but you cant reduce dy/dx .dx to dy by cancelling out algebriacaly dx dy/dx is not a fraction!
YET THIS WHAT SEEMS TO HAVE BEEN DONE HERE!
can anyone explain the theory behind how this was done?
(1 vote)
• It turns out that the dx/dy notation (which was invented by Leibniz) very closely mirrors actual operations. To cancel out the dx's this way is not as simple as cancelling out the fraction. There are actually a number of steps you need to do, but it turns out that S 1/g(y) · dy/dx · dx is equal to S 1/g(y) · dy. So, you are not canceling out fractions, you are applying a rule and Leibniz's notation is so brilliant, that 'canceling out the fraction' actually gives you an expression that is equal.

Disclaimer: I have had this same question for a long time now. I don't actually understand the steps you need to do to show that S 1/g(y) · dy/dx · dx is equal to S 1/g(y) · dy, but I was able to answer your question because I have read this page:
http://math.stackexchange.com/questions/47092/physicists-not-mathematicians-can-multiply-both-sides-with-dx-why
Yeah, I know, it really bugs me to just accept stuff, but I'm gonna have to do the same until I have learned more.
Cheers
• The transition from ln(W-300) to e^(ln(W-300)) was very clever. What suggested that transition, and would it have been acceptable to leave the equation in terms of ln W?