A population is modeled by a function $P$ that satisfies the logistic differential equation $$\frac { d P } { d t } = \frac { P } { 5 } \left( 1 - \frac { P } { 12 } \right) .$$ (a) If $P ( 0 ) = 3$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ? If $P ( 0 ) = 20$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ? (b) If $P ( 0 ) = 3$, for what value of $P$ is the population growing the fastest? (c) A different population is modeled by a function $Y$ that satisfies the separable differential equation $$\frac { d Y } { d t } = \frac { Y } { 5 } \left( 1 - \frac { t } { 12 } \right)$$ Find $Y ( t )$ if $Y ( 0 ) = 3$. (d) For the function $Y$ found in part (c), what is $\lim _ { t \rightarrow \infty } Y ( t )$ ?
A population is modeled by a function $P$ that satisfies the logistic differential equation
$$\frac { d P } { d t } = \frac { P } { 5 } \left( 1 - \frac { P } { 12 } \right) .$$
(a) If $P ( 0 ) = 3$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ?
If $P ( 0 ) = 20$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ?
(b) If $P ( 0 ) = 3$, for what value of $P$ is the population growing the fastest?
(c) A different population is modeled by a function $Y$ that satisfies the separable differential equation
$$\frac { d Y } { d t } = \frac { Y } { 5 } \left( 1 - \frac { t } { 12 } \right)$$
Find $Y ( t )$ if $Y ( 0 ) = 3$.
(d) For the function $Y$ found in part (c), what is $\lim _ { t \rightarrow \infty } Y ( t )$ ?