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Math Help - Family of functions, differential equations

  1. #1
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    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.
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    Quote Originally Posted by cowboys111 View Post
    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.
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    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.
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  4. #4
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    Quote Originally Posted by cowboys111 View Post
    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.
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    Thanks I appriciate it.
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