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Math Help - Another Seperable Differential EQ

  1. #1
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    Another Seperable Differential EQ

    \frac{dz}{dt} + e^{t+z} = 0

    I moved the e^{t+z} to the right side of the equation and came out with.

    \frac{dz}{e^z} = -e^t dt

    Am I right so far, before I take the integral of both sides?


    And can the integral of \frac{1}{e^z} be taken without using parts?
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  2. #2
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    Quote Originally Posted by xwanderingpoetx View Post
    \frac{dz}{dt} + e^{t+z} = 0

    I moved the e^{t+z} to the right side of the equation and came out with.

    \frac{dz}{e^z} = -e^t dt

    Am I right so far, before I take the integral of both sides?


    And can the integral of \frac{1}{e^z} be taken without using parts?
    \frac{dz}{dt} = -e^{t+z}

    \frac{dz}{dt} = -e^t \cdot e^z

    e^{-z} \, dz = -e^t \, dt

    -e^{-z} = C - e^t

    e^{-z} = e^t + C

    -z = \ln(e^t + C)

    z = \ln\left(\frac{1}{e^t + C}\right)
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  3. #3
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    I just have one question, why did you take the reciprocal at the end when you divided out the -1?
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  4. #4
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    Quote Originally Posted by xwanderingpoetx View Post
    I just have one question, why did you take the reciprocal at the end when you divided out the -1?
    power rule for exponents ...

    -\ln{a} = \ln{a^{-1}} = \ln\left(\frac{1}{a}\right)
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  5. #5
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    Wow, I completely blanked on that. Thank you =D
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