Verify that the indicated farrily of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

$\frac{dP}{dt}=P(1-P)$;$,P=\frac{c_1{e}^t}{1c_1{e}^t}$

When $P=\frac{c_1{e}^{\text{'}}}{1c_1{e}^{\text{'}}},$

$\frac{dP}{dt}=$

Thus, in terms of t $,$

$\frac{dP}{dt}-\frac{c_1{e}^t}{1c_1{e}^t}(1-\frac{c_1{e}^t}{1c_1{e}^t})=-\frac{c_1{e}^t}{1c_1{e}^t}(1-\frac{c_1{e}^t}{1c_1{e}^t})$

$=$