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Math Help - Easy First Order ODE

  1. #1
    Member Jason Bourne's Avatar
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    Easy First Order ODE

    Solve,

    y\frac{dy}{dx} = 1+2y

    I know this is easy but I have gone blank its seperable but I had trouble integrating for some reason.

    thanks
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  2. #2
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    \int\frac{y}{1+2y}dy\;=\;\frac{y}{2}-\frac{1}{4}ln|1+2y|+C?
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jason Bourne View Post
    Solve,

    y\frac{dy}{dx} = 1+2y

    I know this is easy but I have gone blank its seperable but I had trouble integrating for some reason.

    thanks
    Notice that
    \frac{y}{1 + 2y} ~dy = dx

    Let u = 2y + 1 \implies du = 2dy so

    \frac{\frac{1}{2}(u - 1) }{u}~\frac{1}{2}du = dx

    \frac{1}{4}\frac{u - 1}{u}~du = dx

    \frac{1}{4} \left ( 1 - \frac{1}{u} \right ) du = dx

    -Dan
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