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Differential equation IVP

  1. Sep 29, 2011 #1
    1. The problem statement, all variables and given/known data
    Find a particular solution to this IVP:

    dy/dx = 1 - 2y
    y(0) = 5/2


    2. The attempt at a solution
    I find -0.5 *ln (1- 2y) = x + C

    However, y = 5/2 gives ln(-4), which is a problem....Have I done something wrong? Any suggestions please? How to find a family of solutions that would be defined for the given point? Thank you.
     
  2. jcsd
  3. Sep 29, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The integral of 1/x is ln|x|, not ln(x)!

    Often the distinction is not important but this is one problem where it is crucial.
     
  4. Sep 29, 2011 #3

    dynamicsolo

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    Homework Helper

    If we look at the differential equation, y' = 1 - 2y , we see that y' = 0 for y = 1/2 . This is what is sometimes called a "stationary solution" (also referred to as the "trivial" solution). If the initial value for y (for any time choice) were 1/2 , the value of y would remain at 1/2 forever. For any initial value y(0) < 1/2 , the function for y will have y' > 0 , with y' becoming smaller as y increases toward 1/2 . By the same token, for any initial value y(0) > 1/2 , y' < 0 , meaning that y will decline toward 1/2 , and at ever slower rates as y approaches that value.

    This results in two cases for the IVP then:

    for y(0) < 1/2 , the integration gives -0.5 ln ( 1 - 2y ) = x + C , which leads to an acceptable value for C because ( 1 - 2y ) is always positive;

    but for y(0) > 1/2 , the integration gives -0.5 ln ( 2y - 1 ) = x + C , which makes the argument of the logarithm positive and maintains the values y will pass through within the domain of the function (and so will give a sensible value for C).

    This is what is contained in the standard result for this problem, y = -0.5 ln | 1 - 2y | = x + C : the presence of the absolute value signals that there are two cases to consider (sadly, a matter often inadequately covered in introductory DE courses).
     
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