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Math Help - Help With differntial equation

  1. #1
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    Question Help With differntial equation

    Please see the scanned image.

    i am newbie to this topic at the moment

    thanks
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  2. #2
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    Hello, master_2m8!

    \frac{dy}{dt} \:=\:\frac{2}{100}y - 50

    We have: . \frac{dy}{dt} - \frac{1}{50}y \:=\:-50

    Integrating factor: . I \:=\:e^{\int\text{-}\frac{1}{50}\,dt} \:=\:e^{\text{-}\frac{1}{50}t}

    Multiply by I\!:\;\;e^{\text{-}\frac{1}{50}t}\frac{dy}{dt} -\frac{1}{50}e^{\text{-}\frac{1}{50}t}y \;=\;-50\,e^{\text{-}\frac{1}{50}t}

    We have: . \frac{d}{dt}\left(e^{\text{-}\frac{1}{50}t}y\right) \:=\: -50\,e^{\text{-}\frac{1}{50}t}

    Integrate: . e^{\text{-}\frac{1}{50}t}y \;=\;-50\!\int e^{\text{-}\frac{1}{50}t}dt

    . . . . . . . . e^{\text{-}\frac{1}{50}t}y \;=\;2500\,e^{\text{-}\frac{1}{50}t} + C


    Multiply by e^{\frac{1}{50}t}\!:\quad\boxed{y \;=\;2500 + C\,e^{\frac{1}{50}t}}

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