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Math Help - calculus - population model/de

  1. #1
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    Exclamation calculus - population model/de

    having so much trouble with de's in general! i think it's a linear de, but really stuck! please help:

    dp/dt = -50 + 15p - p^2

    thanks in advance!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello

    There is a p^2 in the expression so the equation is not linear. However, the two variables p and t can be separated : \frac{\mathrm{d}p}{p^2-15p+50}=-\mathrm{d}t and we'll get the solution by integrating the both sides.

    To achieve this, one needs to make appear a derivative on the LHS. As the denominator contains a p^2, one can try to transform \frac{\mathrm{d}p}{p^2-15p+50} into \mathrm{constant}\cdot\frac{\mathrm{d}u}{1+u^2} which anti-derivative is \mathrm{constant}\cdot \arctan u. Hence solving this problem boils down to finding u. Do you have any idea on how this can be done ?
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  3. #3
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    thanks flyingsquirrel.
    well i have no idea, but i thought maybe i could try 'flipping' it as the rhs is in terms of p

    ie. dt/dp = - 1/(p^2 - 15p + 50)
    and then t = - integral[1/(p^2 - 15p + 50)] dp

    does that work? i'm not sure if that is how to start the question
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