Family of functions, differential equations

**cowboys111** · January 25th 2009, 01:09 PM

Verify that the indicated family of functions is a solution to the given differential equation.

dP/dt=P(1-P); P=(c1*e^t)/(1+c1*e^t)

Do I take the derivative of P first? If so do I just ignore the constants? Thanks for the help.

**Danny** · January 25th 2009, 01:14 PM

Originally Posted by cowboys111

Verify that the indicated family of functions is a solution to the given differential equation.

dP/dt=P(1-P); P=(c1*e^t)/(1+c1*e^t)

Do I take the derivative of P first? If so do I just ignore the constants? Thanks for the help.

No, leave the constants alone. Differentiate P (LHS), substitute P into the RHS and show they are equal.

**cowboys111** · January 25th 2009, 01:36 PM

Im still kind of confused, Is there any way you can show me how you would solve it? My class doesnt start untill tomorrow so I havent had a lecture, Im just trying to read ahead a little bit but my book has horrible examples.

**Danny** · January 25th 2009, 01:51 PM

Originally Posted by cowboys111

Verify that the indicated family of functions is a solution to the given differential equation.

dP/dt=P(1-P); P=(c1*e^t)/(1+c1*e^t)

Do I take the derivative of P first? If so do I just ignore the constants? Thanks for the help.

Sure.

$\frac{dP}{dt} = \frac{c e^t}{\left(1 + c e^t \right)^2}$

$P(1-P) = \frac{c e^t}{1 + c e^t} \left( 1 - \frac{c e^t}{1 + c e^t}\right) = \frac{c e^t}{1 + c e^t} \frac{1}{1 + c e^t} = \frac{c e^t}{\left(1 + c e^t \right)^2}$

see - the same.

**cowboys111** · January 25th 2009, 01:57 PM

Thanks I appriciate it.

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