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Most of Shakespeare's plays are based on stories that were already well known to his audiences. (He never wrote a play about a contemporary subject.) Romeo and Juliet is based on a long narrative poem by Arthur Brooke, which was published in 1562 as The Tragicall Historye of Romeus and Juliet. Brooke's popular poem itself was based on older Italian stories.

Romeo and Juliet, a very young man and a nearly fourteen-year-old girl, fall in love at first sight. They are caught up in an idealized, almost unreal, passionate love. They are in love with love. In his Prologue, Brooke preaches a moral, which people of his time expected. He says that Romeo and Juliet had to die because they broke the laws and married unwisely, against their parents' wishes. But Shakespeare does away with this moralizing. He presents the couple as "star-crossed lovers," doomed to disaster by fate. To understand what starcrossed means, you have to realize that most people of Shakespeare's time believed in astrology. They believed that the course of their lives was partly determined by the hour, day, month, and year of their birth-hence, "the star" under which they were born. But Shakespeare may not have shared this belief. In a later play,

Julius Caesar, Shakespeare has a character question this old idea about astrology and the influence of the stars:

The fault, dear Brutus, is not in our stars, But in ourselves that we are underlings.

Although Shakespeare says in the Prologue that Romeo and Juliet are star-crossed, he does not make them mere victims of fate. Romeo and juliet make decisions that lead to their disaster. More important, other characters have a hand in the play's tragic ending. How important do you think fate is in affecting what happens to us? To what degree do you think we control our own destinies?


A community of hares on an island has a population of 50 when observations begin at t=0. The population for

t0t \geq 0

is modeled by the initial value problem

dPdt=0.08P(1P200)\frac { d P } { d t } = 0.08 P \left( 1 - \frac { P } { 200 } \right)

, P(0)=50. a. Find and graph the solution of the initial value problem. b. What is the steady-state population?


Step 1
1 of 4


Write differential equation in separable form and integrate both sides with respect to tt to obtain general solution in implicit form.

P(t)P(10.005P)dt=0.08dt(1P+1200P)dP=0.08t+ClnPln200P=0.08t+C\begin{align*} \int \dfrac{P'(t)}{P\left( 1-0.005P\right)} \,dt=\int 0.08 \,dt\\ \int \left( \dfrac{1}{P} + \dfrac{1}{200-P}\right) \,dP=0.08t+C\\ \ln \left| P\right| - \ln \left|200-P \right|=0.08t+C\\ \end{align*}

Use initial condition to evaluate constant CC

ln50ln20050=0.080+CC=ln13\begin{align*} \ln \left| 50\right| - \ln \left|200-50 \right|=0.08\cdot 0 +C\\ C=\ln \dfrac{1}{3} \end{align*}

Final solution is

lnPln200P=0.08t+ln13P200P=e0.08t3P(t)=2001+3e0.08t\begin{align*} \ln \left| P\right| - \ln \left|200-P \right|=0.08t+\ln\dfrac{1}{3}\\ \dfrac{P}{200-P}=\dfrac{e^{0.08t}}{3}\\ \boxed{P(t)=\dfrac{200}{1+3e^{-0.08t}}} \end{align*}

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