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Math Help - Another separable differential equation

  1. #1
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    Another separable differential equation

    I understand that this requires integration by parts.

    Find the function (for ) which satisfies the separable differential equation
    with the initial condition .


    I've gotten to the point where

    int y^2 dy = int (10+14x)/x dx

    left side with y is straight forward, but the right side with x is ugly.

    I made u = (10+14x) and dv = 1/x

    du = 14 dx and v = ln(x)

    after this i got something really ugly that i can't even type.

    Please help me.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Why make it so hard? Having \frac{10+14x}x=\frac{10}x+14, and we're done.
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  3. #3
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    yeah, i've tried that
    if I still have to integrate (10/x + 14) by separation of variable
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  4. #4
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    skeeter's Avatar
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    so integrate ...

    \int y^2 \, dy = \int \frac{10}{x} + 14 \, dx

    \frac{y^3}{3} = 10\ln|x| + 14x + C

    y^3 = 30\ln|x| + 42x + C_2

    y = \sqrt[3]{30\ln|x| + 42x + C_2}

    now use your initial condition to determine C_2.
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