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Differential Equations Verifying Implicit Solution

  1. Sep 15, 2011 #1
    I'm given:

    1. [tex]\frac{dX}{dt}=(X-1)(1-2X)[/tex]
    2. [tex]ln(\frac{2X-1}{X-1})=t[/tex]

    and asked to verify that it is an implicit solution to the first order DE given.

    I successfully derived the second equation there to get:

    [tex]\frac{dX}{dt}=\frac{-1}{(2X-1)(X-1)}[/tex]

    So now what? I tried several things and got nowhere.
     
  2. jcsd
  3. Sep 15, 2011 #2

    Mark44

    Staff: Mentor

    Show us the work you did when you differentiated (not derived) the second equation.

    Note that
    [tex]ln(\frac{2X-1}{X-1})= ln(2X - 1) - ln(X - 1)[/tex]

    so that should make differentiation a little easier.
     
  4. Sep 15, 2011 #3
    I did break it up initially:
    [tex]\frac{dx}{dt}=\frac{d}{dt}( ln(2x-1) - ln(x-1))[/tex]
    [tex]\frac{dx}{dt}=\frac{1}{2x-1}(2) - \frac{1}{x-1}(1)=\frac{2}{2x-1} - \frac{1}{x-1}[/tex]
    Then subtracting the fractions:
    [tex]\frac{dx}{dt}= \frac{2(x-1) - (2x-1)}{(2x-1)(x-1)}=\frac{-1}{(2x-1)(x-1)}[/tex]
     
  5. Sep 15, 2011 #4

    Mark44

    Staff: Mentor

    Starting from your second equation, you have
    [tex]t= ln(2x-1) - ln(x-1)[/tex]
    [tex]\Rightarrow \frac{dt}{dt}=\frac{d}{dt}( ln(2x-1) - ln(x-1))[/tex]

    You have to differentiate the right side implicitly, and then solve algebraically for dx/dt. Can you take it from here?
     
  6. Sep 15, 2011 #5
    I solved it! Thanks for the tip.
     
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