# Thread: Family of functions, differential equations

1. ## Family of functions, differential equations

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.

2. 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.

3. 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.

4. 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.

5. Thanks I appriciate it.

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# dp/dt=p(1-p) p=c1e^t/1 c1e^t

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