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Differential Eqns Separable Equation?

  1. Jul 30, 2008 #1
    1. The problem statement, all variables and given/known data

    Problem given basically in the beginning of the book, Sec. 1.4, about separable equations.

    Find general solutions (implicit if necessary, explicit if convenient) of the diff. eq.


    2. Relevant equations

    There was a previous section where they popped this out of thin air:
    A solution of [tex]dy/dx=y^2[/tex] is [tex]y=1/c-x[/tex]

    I'm presuming that that is supposed to be helpful somehow.

    3. The attempt at a solution

    [tex]dy/dx=-2xy^2 => dy/y^2=-2xdx[/tex]

    Now I can't integrate the left side.

    I'm sure that is an abysmal way of going at it. So any help would be appreciated.
  2. jcsd
  3. Jul 30, 2008 #2


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    That's a perfectly good way of going at it. The question is why are you having trouble integrating y-2dy? Didn't you learn a general formula for integrating powers? (Other than -1)
  4. Jul 30, 2008 #3
    I haven't learned a formula for that. I will bet that it involves grabbing the derivative of the denominator from somewhere and thinking of it as a u-substitution but I'm not seeing where that is possible.

    EDIT: Ok I just ignored that y was a function of x and integrated, and it worked out perfectly fine. I am supposing that just this idea is a point of this exercise. When you integrate something in this way, you have an equation and integrate both sides. When you do it that way y is no longer taken as being a function of x. I can accept that as the point, but now why is that possible?
    Last edited: Jul 30, 2008
  5. Jul 30, 2008 #4
    Ah, I think you're looking at it like this [tex]\frac{f'(x)}{[f(x)]^2}[/tex] in your head, and somehow you think of it as a u-substitution where u = f(x).

    Basically the general formula for integrating is

    [tex]\int{x^rdx}= \frac{1}{r+1} x^{r+1}+C[/tex] where [tex]r \neq -1[/tex]
  6. Jul 30, 2008 #5
    Thanks for the reply, konthelion. I was in the process of editing the last post when you posted, I apologize.

    I do know that general rule of integration (it has been severely beaten into my brain) but I was having trouble with the y. My head is saying 'Hey, hold on a minute y is a function of x, you can't do that.' Apparently my head is wrong.
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