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Math Help - separable differential equation

  1. #1
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    separable differential equation

    I was working on my hw and i'm stuck on this question.

    Solve the separable differential equation for ,
    Use the following initial condition:


    [
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  2. #2
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    e^{4u+7t} = e^{4u} \cdot e^{7t}

    separate variables ...

    e^{-4u} \, du = e^{7t} \, dt

    take it from here?
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  3. #3
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    -[ln(-4/7)*7*t-48]/4

    i ended up with this but it's wrong
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  4. #4
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    e^{-4u} \, du = e^{7t} \, dt

    -\frac{1}{4}e^{-4u} = \frac{1}{7}e^{7t} + C

    e^{-4u} = -\frac{4}{7}e^{7t} + C_2

    -4u = \ln\left(C_2 - \frac{4}{7}e^{7t}\right)

    u = -\frac{1}{4}\ln\left(C_2 - \frac{4}{7}e^{7t}\right)

    u(0) = 12

    12 = -\frac{1}{4}\ln\left(C_2 - \frac{4}{7}\right)

    C_2 = e^{-48} + \frac{4}{7}

    u = -\frac{1}{4}\ln\left(e^{-48} + \frac{4}{7} - \frac{4}{7}e^{7t}\right)

    not very pretty.
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