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Homework Help: Straight Forward DE

  1. Jul 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Find 1-parameter family of solutions for the folloing differential equation:

    dr/dβ cot (β) - r = 2

    3. The attempt at a solution
    dr/dβ cot β - r = 2
    dr/β cot β = 2 + r
    dr = (2+r) dβ/cot β
    integral dr/2+r = integral tan β dβ
    ln (2+r) = - ln (cos β) + c

    I take everything to the e power

    2+r = - cos β + e^c
    r = c cos - 2

    While the book has r = c sec - 2. I suspect this might be because of taking e^- ln, if it is, please explain why that happens.
  2. jcsd
  3. Jul 9, 2012 #2


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    Science Advisor

    You have errors when taking e to the power of each side. (NOT "to the e power"!)
    [itex]e^{-ln(cos(\beta))+ C}[/itex] is NOT "[itex]-cos(\beta)+ e^c[/itex]" for two reasons. First, [itex]e^{-ln(A)}= e^{ln(1/A)}= 1/A[/itex], not -A. Second [itex]e^{A+ B}= e^Ae^B[/itex], not [itex]e^A+ e^B[/itex].
    [tex]e^{-ln(cos(\beta))+ c}= e^{ln(1/cos(\beta)}e^c= C/cos(\beta)= C sec(\beta)[/tex]
    where [itex]C= e^c[/itex].
    Last edited by a moderator: Jul 9, 2012
  4. Jul 9, 2012 #3
    Sorry, I was sloppy with my wording hehe.

    What i did was [itex]e^A+e^B[/itex], then I incorporated the constant when I wrote [itex]c cos[/itex]

    Many thanks!
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