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

1. ## 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.

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

Originally Posted by aew1324657
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.

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

Originally Posted by Prove It
\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