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Finish this problem? (Diff Eq)

  1. Sep 18, 2013 #1
    1. The problem statement, all variables and given/known data
    [tex]\frac{dx}{dt}=x-x^{2}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    I think the only thing I have wrong so far is how to finish it because I cant find anything wrong with my work, but I don't know how the book gets their final answer.
    Separate variables...
    [tex]\int \frac{dx}{x(1-x)}=\int dt[/tex]
    Partial fraction decomposition..
    [tex]\int (\frac{1}{x}-\frac{1}{x-1})dx=t+C\\ln|x|-ln|x-1|=t+C\\ln|\frac{x}{x-1}|=t+C[/tex]
    Using properties of e...
    [tex]\frac{x}{x-1}=e^{t}e^{C}[/tex]
    D=e^C (professor wants us to write it like this) and multiply by x-1 on both sides..
    [tex]x=De^{t}(x-1)[/tex]

    That's not the solution though and I'm not sure why.
    Final answer: [tex]x(t)=\frac{C}{C-e^{t}}[/tex]
     
  2. jcsd
  3. Sep 18, 2013 #2

    ehild

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    Gold Member

    You have to isolate x, writing the solution in the form x(t)=f(t). Expand your equation, collect the terms with x at one side, factor it ....

    The final solution you quoted is not correct. There must be a minus in front of t in the exponent.

    ehild
     
  4. Sep 19, 2013 #3

    verty

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    Homework Helper

    It was 95% complete, just to collect terms.
     
  5. Sep 19, 2013 #4
    I don't know why I didn't see that before... Maybe I was just too tired.. Thank you.
     
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