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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due July 3 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2018 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: The Wronskian, affine spaces, and linear ODEs}
\end{center}


%\noindent Read the following:
%\begin{itemize}
%\item Chapters 1 \& 2 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
%\item VGT Exercises 1.1--1.4, 1.8--1.12, 2.13--2.17. \vspace{-2mm}
%\item The article \emph{Group Think} by Steven Strogatz, which
%  appeared in the NY Times in 2010. \vspace{-2mm}
%\item Francis Su's essay on good mathematical writing.
%\end{itemize} 

\begin{enumerate}


\item Solve the following ODEs:
  \begin{multicols}{2}
  \begin{enumerate}
    \item $y'+9y=0$ 
    \item $y''+9y=0$
    \item $y''+5y'+4y=0$ 
    \item $4y''+4y'+5y=0$ 
    \item $y'+9y=18$ 
    \item $y''+9y=e^{3t}$
    \item $y''+5y'+4y=e^t$ 
    \item $4y''+4y'+5y=e^{2t}$ \bigskip
  \end{enumerate}
  \end{multicols}



\item The function $y_1(t)=t$ is a solution of $t^2y''+5ty'-5y=0$.
  \begin{enumerate}
    \item Use the \emph{reduction of order} technique to find a second
      solution, $y_2(t)$.
    \item Compute the Wronskian of $y_1(t)$ and $y_2(t)$. \bigskip
  \end{enumerate}



\item Let $v_1$ and $v_2$ be two linearly independent vectors in $\R^3$,
  i.e., they determine a plane $P$. 
  \begin{enumerate}
    \item Sketch the plane $P=\{C_1v_1+C_2 v_2\mid C_1,C_2\in\R\}$ in
      $\R^3$.
    \item Sketch the vectors $w_1:=\frac{1}{2}v_1+\frac{1}{2}v_2$ and
      $w_2:=\frac{1}{2}v_1-\frac{1}{2}v_2$. Do these ``determine'' the
      same plane?  In other words, is the following (infinite) set of
      vectors
      \[
      \left\{C_1\left(\tfrac{1}{2}v_1+\tfrac{1}{2}v_2\right)
      +C_2\left(\tfrac{1}{2}v_1-\tfrac{1}{2}v_2\right) \mid C_1,C_2\in\R\right\}
      \]
      the same as $\{C_1v_1+C_2 v_2\mid C_1,C_2\in\R\}$?
    \item Is $\{v_1,v_2\}$ a basis for $P$? Is $\{w_1,w_2\}$ a basis
      for $P$? Why or why not?
    \item Calculate the determinant of the matrix
      $\left[\!\!\begin{array}{cc}1/2 & 1/2 \\ 1/2 &
      -1/2\end{array}\!\!\right]$. Is this matrix invertible? 
      How does this relate to HW 1, Problem \#4? \bigskip
  \end{enumerate}



\item In this problem, we will understand from a linear algebra
  perspective why we can write solutions to ODEs for simple harmonic
  motion using complex exponentials or sines and cosines.
  \begin{enumerate}
  \item Consider the ODE $y''-\omega^2y=0$. If we assume that
    $y(t)=e^{rt}$ and plug this back in and solve for $r$, we get that
    $r=\pm\omega$. Therefore, the general solution is
    $y(t)=C_1e^{\omega t}+C_2e^{-\omega t}$, i.e., $\{e^{\omega
    t},e^{-\omega t}\}$ is a \emph{basis} for the solution
    space. Using the fact that $e^{\omega t}=\cosh\omega t+\sinh\omega
    t$, find a basis for the solution space involving hyperbolic trig
    functions, and write the general solution using these
    functions. \emph{Hint}: Look at the previous problem!
  \item Repeat the above exercise for the ODE
    $y''+\omega^2y=0$. Specifically, if we assume that $y(t)=e^{rt}$,
    then we get that $r=\pm i\omega$. Therefore, the general solution
    is $y(t)=C_1e^{i\omega t}+C_2e^{-i\omega t}$, i.e., $\{e^{i\omega
      t},e^{-i\omega t}\}$ is a basis for the solution space. Using
    Euler's formula: $e^{i\omega t}=\cos\omega t+i\sin \omega t$, find
    a basis for the solution space involving sine and cosines, and
    write the general solution using these functions. \bigskip
  \end{enumerate}


\item Suppose that the temperature of glass of water sitting outside
  obeys Newton's law of cooling. It is reasonable to expect that the
  ambient temperature is not constant, but rather a function $A(t)$ of
  time. Moreover, it is reasonable to assume that the ambient
  temperature is sinusoidal, so let $A(t)=B\sin\omega t$, where $B$ is
  some constant. We now have the following equation that we wish to
  solve:
  \[
  T'=k(B\sin\omega t-T)\,.
  \]
  \begin{enumerate}
  \item Solve the related homogeneous equation, $T_h'=-kT_h$.
  \item To solve the ODE, we need to find any particular solution, so
    assume there is one of the form $T_p(t)=a\cos\omega t+b\sin\omega
    t$. Plug this back into the ODE, equate coefficients of the sine
    and cosine terms, and solve for $a$ and $b$ in terms of the
    amplitude $B$, the frequency $\omega$, and the constant $k$.
  \item Find the general solution to this ODE.
  \item Give a qualitative physical description of what the particular
    solution $T_p(t)$ represents, and why. [Hint: Consider the
      long-term behavior of the temperature $T(t)$.]
  \end{enumerate}

\end{enumerate}

\end{document}
