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

• Aug 10th 2012, 09:58 PM
aew1324657
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.
• Aug 10th 2012, 10:29 PM
Prove It
Re: proof to a differential equation involving hyperbolic identities cosh and sinh.
Quote:

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.
• Aug 11th 2012, 03:59 AM
aew1324657
Re: proof to a differential equation involving hyperbolic identities cosh and sinh.
Quote:

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