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General Integration Question

  1. Jul 26, 2008 #1
    I solved a pretty routine first-order diff. eq. where you simply separate the variables.

    xcos(x)(dy/dx) - sin(y) = 0

    => [tex]\int cot(y)dy[/tex] = [tex]\int dx/x[/tex]

    Now, I thought that you would get an arbitrary constant, C, on both sides and they would cancel each other out, but that's wrong. My book lets e^C = A (why?).

    The answer should be sin(y) = Ax, but I didn't get that because I cancelled out the constant. I suppose my question is why does this happen?
     
  2. jcsd
  3. Jul 26, 2008 #2

    Defennder

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    The constant C is arbitrary, which means to say it doesn't have a fixed unknown value. Therefore you can't assume that they have the same value on both sides and it cancel each other out.
     
  4. Jul 27, 2008 #3
    So I could set the LHS constant to be C_1 and the RHS to be C_2 then their difference can be a new constant A?
     
  5. Jul 27, 2008 #4

    tiny-tim

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    Hi Oneiromancy! :smile:
    yes … except

    i] it's A = eC1-C2

    ii] the examiners will expect you to take the short-cut, and just write one C, on one side of the equation, rather than write two and subtract. :smile:
     
  6. Jul 27, 2008 #5

    HallsofIvy

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    The reason the constant is multiplied is that direction integration
    gives you ln(sin(y))= ln(x)+ C and then taking the exponential of both sides,
    [tex]e^{ln(sin(y))}= e^{ln(x)+ C}[/tex]
    [tex]sin(y)= e^{ln(x)}e^C= Ax[/itex]
    where A= eC.
     
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