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Math Help - When more than one particular solution fits the initial conditions

  1. #1
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    When more than one particular solution fits the initial conditions

    So I've got a general solution to a differential equation:

    y = x/4 (ln x + D)^2

    and I'm to solve it for the general initial value problem: y(x0) = y0, x>0, y>0

    I then solved the equation for D, getting two results:

    D = -ln (x0) + 2(y0/x0)^(1/2)

    and

    D = -ln (x0) - 2(y0/x0)^(1/2)

    My problem now is that I have no idea how to pick one solution for D over the other. Both fit all restrictions as far as I can tell. Is it possible that both are valid and that I would need further restrictions to choose one of them?
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: When more than one particular solution fits the initial conditions

    Both solutions are correct, meaning that with either value of D the equation meets the condition that y(x_0) = y_0. I have attached a plot showing the two solutions for (X_0,Y_0) = (1,1) - note that they both pass through (1,1). You can't pick one over the other without introducing some other constraint, such as the slope at x_0 or the value for y at another point.

    When more than one particular solution fits the initial conditions-plots.jpg
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