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\title{Applied Differential Equations and Modeling}
\author{Homework 3}
\date{Due in class Thursday, March 1, 2018}
\maketitle

\begin{enumerate}

\item For each of the following initial value problems, find the
solution and state how the interval of existence of the solution
depends on the initial value $y_0$.
\begin{enumerate}
\item $\displaystyle y' = -4 \, \frac{t}y \,, \qquad y(0) = y_0$
\item $\displaystyle y' + y^3 = 0 \,, \qquad y(0) = y_0$
\end{enumerate}

\item Sometimes it is possible to solve a nonlinear equation by making
a change of the dependent variable that converts it into a linear
equation. The most important such equation has the form 
\[
  y' + p(t) \, y = q(t) \, y^n 
\]
and is called Bernoulli's equation.
\begin{enumerate}
\item Show that if $n \neq 0, 1$, then the substitution $v = y^{1-n}$
reduces Bernoulli's equation to a linear equation.

\item Use this method to solve the initial value problem
\[
  t^2 \, y' + 2t \, y - y^3 = 0 \,, \quad y(1)=1 \,.
\]

\item Use this method to find, by a different method than the one
explained in class, the solution to the logistic differential equation
\[
  y' = r \, y \, \Bigl( 1 - \frac{y}K \Bigr) \,, \quad
  y(0) = y_0 \,.
\]
\end{enumerate}

\item For each of the following equations, determine the equilibrium
points where $y'(x)=0$ and classify each as stable ($y'$ changes sign
from positive to negative at $x$) or unstable ($y'$ changes sign from
negative to positive at $x$).  Sketch a few solution curves (without
trying to solve the equation) in the $t$-$y$ plane.
\begin{enumerate}
\item $\displaystyle y' = y - y^2$
\item $\displaystyle y' = y \, (y-1) \, (y-2)$
\item $\displaystyle y' = \e^y - 1$
\end{enumerate}

\item Let $P(t)$ denote the number of fish in a lake at time $t$, and
  let $C$ denote the ``carrying capacity'' of the lake. Suppose
  further that fishermen catch a fraction $k$ of the fish per unit
of time, so that the population satisfies the equation
\[
  \frac{dP}{dt} = \Bigl( 1 - \frac{P}{C} \Bigr) \, P - k \, P 
  \qquad \text{with} \qquad
  P(0) = P_0 \,.
\]
\begin{enumerate}
\item For given values of $C$ and $k$, when is the population
  increasing, when is it decreasing?
\item For given values of $C$ and $k$, how many fish will be in the
  lake in the long run?  
(You do not need to solve the differential equation to answer this
question!) 
\item Which value of the  fishing rate $k$ maximizes the number of
  fish caught in the long run?
\item Solve the differential equation explicitly and check that your
  solution is
  consistent with your answer to part (b).
\end{enumerate}

\end{enumerate}
\end{document}
