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\title{Applied Differential Equations and Modeling}
\author{Homework 1}
\date{Due in class Monday, February 11, 2019}
\maketitle

\begin{enumerate}

\item For each of the following, 
\begin{enumerate}
\item Draw a direction field for the given differential equation
\item Based on the inspection of the direction field, describe how
solutions behave for large $t$.
\item Find the general solution of the given differential equation,
and use it to determine how solutions behave as $t \to \infty$.
\end{enumerate}
\begin{gather}
  2 y' + y = 3t \\
  t y ' - y = t^2 \e^{-t}
\end{gather}
\item Find the solution of the given initial value problem.
\begin{enumerate}
\item $y' - y = 2t \e^{2t}$, \quad $y(0)=1$
\item $t y' + 2y = t^2 - t + 1$, \quad $y(1) = \tfrac12$, \quad $t>0$
\item $t y' + 2 y = \sin t$, \quad $y(\pi/2) = 1$, \quad $t>0$
\item $y' = (\e^{-x} - \e^x)/(3+4y)$, \quad $y(0)=1$
\end{enumerate}


\item In each of the following problems, find the critical value for
the initial value $a$ where the solution changes from going to
$-\infty$ as $t \to \infty$ to going to $\infty$ as $t \to \infty$.
\begin{enumerate}
\item $y' - \tfrac12 y = 2 \cos t$, \quad $y(0) = a$
\item $2y' - y = \e^{t/3}$, \quad $y(0) = a$
\end{enumerate}

\item Solve the initial value problem
\[
  y' = 2 y^2 + x y^2, \quad y(0)=1
\]
and determine where the solution attains its minimum value.
\end{enumerate}

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