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Math Help - Seperation of Variables (inc cos and exp)

  1. #1
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    Seperation of Variables (inc cos and exp)

    Hi all,
    Problem below:

    Solve the following differential equation

     \frac{dy}{dx} = \frac{cos(2x+1) + e^{3x}}{y}

    I can re arrange this into:

    \frac{dy}{y} = cos(2x+1) + e^{3x}dx

    From there integration brings:

    \ln|y| + c = \frac{sin(2x+1)}{2} + \frac{e^{3x}}{3} + c

    then multiplying by e^{x}:

    y + c = e^{\frac{sin(2x+1)}{2} + \frac{e^{3x}}{3} + c}

    Now I'm stuck. I can't see any way to simplify the e^{()} value on the right. It doesn't really feel solved to me... Is it?
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  2. #2
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    Lexington, MA (USA)
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    Hello, isp_of_doom!

    Your first step is wrong . . .


    \frac{dy}{dx} \:=\: \frac{\cos(2x+1) + e^{3x}}{y}

    We have: . y\,dy \;=\;\left[\cos(2x+1) + e^{3x}\right]\,dx

    Integrate: . \tfrac{1}{2}y^2 \;=\;\tfrac{1}{2}\sin(2x+1) + \tfrac{1}{3}e^{3x} + C

    . . . . . . . . . y^2 \;=\;\sin(2x+1) + \tfrac{2}{3}e^{3x} + C

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