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Math Help - Implicit Form

  1. #1
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    Implicit Form

    Completly stuck and do not have a clue, please help! I need to find in implicit form the general solution of the differential equation:

    \frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}

    This is then followed by finding the corresponding particular solution (in implicit form) that satisfies the initial condition:

    y=1/6 when x=0

    Many thanks
    Last edited by mr fantastic; July 14th 2009 at 03:19 PM. Reason: Fixed the latex and equation
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  2. #2
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    \frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}

    \frac{dy}{y^2} = 3e^{-2x}\sqrt{8+e^{-2x}} \, dx<br />

    -\frac{1}{y} = -\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C

    can you finish up?
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  3. #3
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    Reply to your solution

    Hi thanks for your reply, still being emotional about this!

    I've come up with this solution to finish it, could you please check it and if it's wrong please show me where!

    From where you left me:

    8y dy/dx = (e^-2x)^3/2dx

    y8=2/x-c

    y=4/3 x=0

    Many thanks
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  4. #4
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    \frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}

    \frac{dy}{y^2} = 3e^{-2x}\sqrt{8+e^{-2x}} \, dx<br />

    -\frac{1}{y} = -\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C

    y = \frac{1}{\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C}

    y(0) = \frac{1}{6}

    \frac{1}{6} = \frac{1}{27 + C}

    C = -21

    y = \frac{1}{\left(8 + e^{-2x}\right)^{\frac{3}{2}} - 21}
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