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Solving a first order differential equation

  1. Aug 11, 2012 #1

    the differential equation i am attempting to solve is:

    [tex] \frac {dP} {dx} = \frac {gP} {1+P/Psat} [/tex]

    here is what I have done:

    [tex] \frac {dP} {dx} = \frac {gP*Psat} {Psat+P} [/tex]

    divide both sides by [tex] \frac {Psat+P} {gP*Psat} [/tex]

    to get:
    [tex] \frac {Psat+P} {P*Psat} \frac {dP} {dx} =g [/tex]

    [tex]\int \frac {Psat+P} {P*Psat} dp = \int gdx [/tex]

    [tex]\int \frac {dp} {P}+ \int \frac {dp} {Psat} =gx+c [/tex]
    [tex]ln(P)+ \frac {P} {Psat} =gx+c [/tex]

    now how do i rearrange P on one side with everything else on the other side
  2. jcsd
  3. Aug 11, 2012 #2
    By realizing, P = ln(eP)
  4. Aug 11, 2012 #3
    Hi zak8000 !

    In practice, the equation ln(P)+P/Psat = gx+e is solved thanks to numerical methods.
    The analytic solution requieres a special function W(X), namely the Lambert W function.
    P/Psat = W(X) with X=exp(gx+e)/Psat
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