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Integrating Factor ODE Fix

  1. Sep 10, 2006 #1
    Got the eqn dy/dx=x(1-y) and it can be solved both linear and separable methods.(Linear method being using a integrating factor) Problem im having is that with this two methods i get two different (yet similar answers) and was wondering if you can see my problem with this two methods im using.

    Integrating Factor method:
    y'+xy=x, u'(x)=e^(x^2/2)

    [e^(x^2/2)y]'=x*e^(x^2/2)

    e^(x^2/2)y=integral(x*e^(x^2/2)), do u substitution, get...

    e^(x^2/2)y=e^(x^2/2)+c

    y=1+c/e^(x^2/2) or y=1+c*e^(-x^2/2)

    Separable method:
    dy/(1-y)=x dx, integrate both sides

    -ln(1-y)=e^(x^2/2)+C, raise both sides to e.

    1/(1-y)=K*e^(x^2/2)+C, rearrange to get y=.

    y=1-1/K*e^(x^2/2)

    so we get two different answers with these methods, where is the problem lieing or are both wrong?
     
  2. jcsd
  3. Sep 11, 2006 #2

    Galileo

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    You got a bit sloppy near the end. Some mistakes are just typo's I think.

    [tex]-\ln(1-y)=\frac{1}{2}x^2+C[/tex]
    [tex]1-y=K\exp(-\frac{1}{2}x^2)[/tex]
    [tex]y=1-K\exp(-\frac{1}{2}x^2)[/tex]

    So it's the same.
     
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