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Verify an expression is an implicit solution to a first order DE.

  1. Aug 31, 2012 #1
    1. The problem statement, all variables and given/known data
    Verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution y=∅(x) in each case.


    2. Relevant equations

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

    3. The attempt at a solution
    I know I need to solve the 't=' portion for X and here it is:

    ln([itex]\frac{2X-1}{X-1}[/itex])=t =>

    [itex]\frac{e^t-1}{e^t-2}[/itex]=X

    In -explicit- cases I would differentiate X to find X' and replace the original X' from the given differential equation to see if it works out to 0.

    I don't know how to go about proving it in the implicit case and everything I'm finding when I google is how to do implicit differentiation from a calc 2-3 perspective.

    Thank you
     
  2. jcsd
  3. Aug 31, 2012 #2

    Bacle2

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    I don't see how you go from:

    ln ( (2X-1)/(X-1))=t to (et-1)/(et -2) =X


    I would think of raising e to both sides.
     
  4. Aug 31, 2012 #3
    Exactly. When you do that the ln goes away and you end up with e^t on the other side. We do some algebra and end up with the above result. I have verified this with wolframalpha.

    Edit: Here are the steps for the curious-

    Exponentiate both sides to get: (2X-1)/(X-1)=e^t
    Multiply both sides by the denominator (x-1) to get: 2x-1=xe^t-e^t
    Get the components with x's in them together: xe^t-2x=e^t-1
    Factor out the x: x(e^t-2)=e^t-1
    Divide both sides by (e^t-2) to get x by itself: x=(e^t-1)/(e^t-2)

    Edit 2:
    The back of the book says that x=(e^t-1)/(e^t-2) is a solution. I have to figure out why that's the case.
     
    Last edited: Aug 31, 2012
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