$(Challenging)$ Then the logistic differential equation $is$

$\frac{dP}{(d)t}=rP(1-\frac{P}{K})$

The function $P(t)$ represents the population $of$ this organism $as$ a function

$of$ time $t.$ Let $K$ represent the carrying capacity for a particular organism

$in$ a given environment, and let $rbe$ a real number that represents the

growth rate$(The$ carrying capacity $ofan$ organism $in$ a given environment

$is$ defined $tobe$ the maximum population $of$ that organism that the $en-$

vironment can sustain indefinitely$).$ For simplicity, put $r=1,K=1,$ try

$to$ solve the following logistic differential equation:

$\frac{dP}{(d)t}=P(1-P)$