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\begin{document}

\title{Integration Workshop 2020 \\-- Ordinary Differential Equations --\\ Problems}
\author{Colin Clark}
\date{August 8, 2020}
\maketitle

\section*{Existence \& Uniqueness of Solutions via Picard Iteration}
\begin{enumerate}
  \item Repeat the steps in Example 2 with the equation \(y'(t) = -y\), with \(y_0 = 1\).
  \item Repeat the steps in Example 2 with the equation \(y'(t) = -y\), with \(y_0 = 1\).
  \item Discuss whether you could repeat the steps in Example 2 with the equation \(ty'(t) = y-1\).
  \item Discuss whether you could repeat the steps in Example 2 with the equation \(y'(t) = y^{1/3} + t\).
  % \item Verify that \(y'(x) = \sqrt{|x|}\)
  % \item
\end{enumerate}
\newpage

\section*{Systems of Linear Equations}
\begin{enumerate}[resume]
  \item Express \(e^{At}\) by its Taylor Series to verify that \(\bm{y} = e^{At} \bm{y_0}\) satisfies the differential equation \(\bm{y}' = A\bm{y}\) subject to \(\bm{y} = \bm{y}_0\) at \(t = 0\). The Taylor Series converges for all  \(\bm{R} \in \mathbb{R}^2\), so you may differentiate term-by-term.
  \item Compute \(e^A\) for
  \begin{enumerate}
    \item \(A = \begin{bmatrix} 1 &  2 \\ 3 & 2 \end{bmatrix} \)
    \item \(A = \begin{bmatrix*}[r] 4 & -1 \\ 1 & 2 \end{bmatrix*} \)
  \end{enumerate}
  \item Sketch phase planes for the following systems of differential equations
  \begin{enumerate}
    \item \(\bm{y}' = \begin{bmatrix*}[r] 1 & -2 \\ -2 & 1\end{bmatrix*} \bm{y} \quad \left(= \begin{bmatrix*}[r]  1& 1\\ -1 & 1 \end{bmatrix*}\; \begin{bmatrix*}[r] 3 & 0 \\ 0 & -1\end{bmatrix*}\; \begin{bmatrix*}[r]  1& 1\\ -1 & 1 \end{bmatrix*}^{-1} \bm{y} \right)\)
    \item \(\bm{y}' = \begin{bmatrix*}[r] 1 & -2 \\ 1 & -2 \end{bmatrix*} \bm{y}  \quad \quad\left( =\begin{bmatrix*}[r] \phantom{-}1 & 2\\ 1 &  1\end{bmatrix*}\; \begin{bmatrix*}[r] -1 & 0 \\ 0 & 0\end{bmatrix*}\; \begin{bmatrix*}[r] \phantom{-}1 & 2\\ 1 & 1 \end{bmatrix*}^{-1} \bm{y} \right)\)
    \item \(\bm{y}' = \begin{bmatrix*}[r] 1 & -2 \\ 2 & 1 \end{bmatrix*} \bm{y}  \quad \quad \left(= \begin{bmatrix} i & -i\\ 1 & -1 \end{bmatrix}\; \begin{bmatrix}1 + 2i & 0 \\ 0 & 1 - 2i\end{bmatrix}\; \begin{bmatrix} i & -i\\ 1 & -1 \end{bmatrix}^{-1} \bm{y}\right)\)
  \end{enumerate}
  \item Sketch the phase plane for the following system of differential equations
  \begin{align*}
    \dot{x}(t) = -y(t)\\
    \dot{y}(t) = \phantom{-}x(t)\\
    \dot{z}(t) = -z(t)
  \end{align*}
\end{enumerate}
\newpage

\section*{Techniques for Explicit Solutions}
\begin{enumerate}[resume]
  \item Find the general solution to
  \begin{equation}
    \frac{dP}{dt} = k P\left(1- \frac{P}{K}\right)
  \end{equation}
  \item Find the the general solutions to the following differential equations
  \begin{enumerate}
    \item \( y'' +3 y' -10 y= 0\)
    \item \(\ddot{x} + 4\dot{x} + 4x  = 0\)
  \end{enumerate}
  \item Find the general solutions to the following differential equations
  \begin{enumerate}
    \item \(s^2 y'' - 2 s\dot{y} + 2 y = 0\)
    \item \(t^2 \ddot{x} + 3t\dot{x} + 2x = 0\)
  \end{enumerate}
  \item Solve \begin{equation} s \frac{dy}{ds} + (s+1)y = 3\end{equation}
  \item Use the method of the variation of parameters so solve the following differential equations
  \begin{enumerate}
    \item \(s^2 y'' - 2 s y'+ 2 y = s\)
    \item \(\ddot{x} + 4\dot{x} + 4x  = \cosh{t} \)
  \end{enumerate}
\end{enumerate}

\end{document}
