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Math Help - 2nd order ODE

  1. #1
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    2nd order ODE

    Hi,

    I'm having issues with this (I'm sure very basic) example:

    y'' - y' + 2y^2(1 - y) = 0

    Find a value of the constant a s.t. y = (1 + e^(ax))^-1 is a solution of this equation.

    I assumed I would be able to work out y'' and y', plug into the equation and solve for a but this doesn't seem to work.

    Any ideas please?
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  2. #2
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    Quote Originally Posted by s0791264 View Post
    Hi,

    I'm having issues with this (I'm sure very basic) example:

    y'' - y' + 2y^2(1 - y) = 0

    Find a value of the constant a s.t. y = (1 + e^(ax))^-1 is a solution of this equation.

    I assumed I would be able to work out y'' and y', plug into the equation and solve for a but this doesn't seem to work.

    Any ideas please?
    Please show your working so that it can be reviewed.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Please show your working so that it can be reviewed.
    y = (1 + e^(ax))^-1
    y' = -ae^(ax)(1 + e^(ax))^-2
    y'' = -a(^2)e^(ax)(1 + e^(ax))^-2 + 2(a^2)e^(2ax)(1 + e^(ax))^-3

    Plugging this into the equation I get

    -a(^2)e^(ax)(1 + e^(ax))^-2 + 2(a^2)e^(2ax)(1 + e^(ax))^-3 + ae^(ax)(1 + e^(ax))^-2 + 2(1 + e^(ax))^-2(1 - (1 + e^(ax))^-1) = 0

    Sorry about the clumsiness.

    I have tried different ways of manipulating this but I keep ending up with something like

    e^(ax)(-(a^2) + a + 2) + e^(2ax)(a^2 + a) = 0

    which I don't know how to solve.
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  4. #4
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    Quote Originally Posted by s0791264 View Post
    y = (1 + e^(ax))^-1
    y' = -ae^(ax)(1 + e^(ax))^-2
    y'' = -a(^2)e^(ax)(1 + e^(ax))^-2 + 2(a^2)e^(2ax)(1 + e^(ax))^-3

    Plugging this into the equation I get

    -a(^2)e^(ax)(1 + e^(ax))^-2 + 2(a^2)e^(2ax)(1 + e^(ax))^-3 + ae^(ax)(1 + e^(ax))^-2 + 2(1 + e^(ax))^-2(1 - (1 + e^(ax))^-1) = 0

    Sorry about the clumsiness.

    I have tried different ways of manipulating this but I keep ending up with something like

    e^(ax)(-(a^2) + a + 2) + e^(2ax)(a^2 + a) = 0

    which I don't know how to solve.
    Not sure where your mistake is but this is what I got after simplying

     <br />
\frac{(a+1)e^{ax}(ae^{ax}-a+2)}{\left(e^{ax}+1\right)^3}.<br />
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