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Math Help - proof to a differential equation involving hyperbolic identities cosh and sinh.

  1. #1
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    proof to a differential equation involving hyperbolic identities cosh and sinh.

    Hi, really struggling with this and a worked solution to follow would be very very much appreciated.. I can't seem to get my head around this

    "Consider the differential equation dp/dt=p(2000-p)/2000 with the initial condition p(0)=500.
    Solve this differential equation and prove that p(t)=2000/(3cosh(t)-3sinh(t)+1."

    thanks.
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  2. #2
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    Re: proof to a differential equation involving hyperbolic identities cosh and sinh.

    Quote Originally Posted by aew1324657 View Post
    Hi, really struggling with this and a worked solution to follow would be very very much appreciated.. I can't seem to get my head around this

    "Consider the differential equation dp/dt=p(2000-p)/2000 with the initial condition p(0)=500.
    Solve this differential equation and prove that p(t)=2000/(3cosh(t)-3sinh(t)+1."

    thanks.
    \displaystyle \begin{align*} \frac{dp}{dt} &= \frac{p(2000-p)}{2000} \\ \frac{1}{p(2000 - p)}\,\frac{dp}{dt} &= \frac{1}{2000} \\ \int{\frac{1}{p(2000-p)}\,\frac{dp}{dt}\,dt} &= \int{\frac{1}{2000}\,dt} \\ \int{\frac{1}{p(2000-p)}\,dp} &= \int{\frac{1}{2000}\,dt} \end{align*}

    The integral on the left can be solved using Partial Fractions.
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    Re: proof to a differential equation involving hyperbolic identities cosh and sinh.

    Quote Originally Posted by Prove It View Post
    \displaystyle \begin{align*} \frac{dp}{dt} &= \frac{p(2000-p)}{2000} \\ \frac{1}{p(2000 - p)}\,\frac{dp}{dt} &= \frac{1}{2000} \\ \int{\frac{1}{p(2000-p)}\,\frac{dp}{dt}\,dt} &= \int{\frac{1}{2000}\,dt} \\ \int{\frac{1}{p(2000-p)}\,dp} &= \int{\frac{1}{2000}\,dt} \end{align*}

    The integral on the left can be solved using Partial Fractions.
    Thanks heaps i got it
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