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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due February 7
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 434 $|$
    Advanced Engineering Mathematics $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

 \item 
   \begin{enumerate}
   \item Consider an ODE
     \[
     z''+a(t)z'+b(t)z=0\,,
     \]
     and suppose that you know one complex-valued solution,
     $z(t)=x(t)+iy(t)$, where $x(t)$ and $y(t)$ are real-valued
     functions. Substitute $z(t)$ into the above equation and equate
     real and imaginary parts to show that $x(t)$ and $y(t)$ are also
     solutions. In what case do you have enough information to
     determine the general solution. What is the general solution in
     this case?
   \item Now, consider the ODE $z''+6z'+10z=0$. Find a solution of the
     form $z_1(t)=e^{rt}$ for some $r$, and use Euler's equation,
     $e^{i\theta}=\cos\theta+i\sin\theta$, to break this solution into
     its real and imaginary parts.
   \item Use the result of Part~(a) to write the general solution.
   \end{enumerate}
     

\item Consider a 2nd order inhomogeneous ODE
  \[
  y''+a(t)y'+b(t)y=g(t)\,.
  \]
  \begin{enumerate}
  \item First, consider the associated homogeneous equation,
    \[
    y''_h+a(t)y'_h+b(t)y_h=0.
    \]
    Verify that the set of solutions is a vector space by showing that
    if $y_1(t)$ and $y_2(t)$ are any two solutions, then
    $C_1y_1(t)+C_2y_2(t)$ is a solution for any $C_1,C_2\in\R$. [Note
      that this shows closure under addition and scalar multiplication
      at the same time!]
    \item Now, consider the original inhomogeneous ODE, and suppose
      that $y(t)$ is the general solution and $y_p(t)$ is \emph{any}
      one particular solution. Show by direct substitution that
      $y(t)-y_p(t)$ solves the homogeneous equation.
    \item Conclude that $y(t)=y_h(t)+y_p(t)$, where $y_h(t)$ is the
      general solution to the related homogeneous equation. Given the
      extra knowledge that $y_h(t)$ is a $2$-dimensional vector space,
      this allows us to conclude that
      $y(t)=C_1y_1(t)+C_2y_2(t)+y_p(t)$. Summarize this in a sentence
      (i.e., explain precisely what $y_1(t)$ and $y_2(t)$ are, because
      not just any two functions work).
  \end{enumerate}


\item Solve the following linear, homogeneous ODEs.
\begin{enumerate}
\item $y'+3y=0$ 
\item $4y''+4y'+5y=0$
\item $y''-6y'+9y=0$
\item $y''+5y'+4y=0$
\end{enumerate}


\item Solve the following linear, inhomogeneous ODEs. Feel free to use
  the results of the previous problem without re-deriving them.
\begin{enumerate}
\item $y'+3y=2t^2$ 
\item $4y''+4y'+5y=e^{3t}+\sin 2t+10$
\item $y''-6y'+9y=e^{2t}$
\item $y''+5y'+4y=e^{-t}$
\end{enumerate}


% Polking 4.5 #30
\item If $y_f(t)$ is a solution of
  \[
  y''+a(t)y'+b(t)y=f(t)
  \]
  and $y_g(t)$ is a solution of
  \[
  y''+a(t)y'+b(t)y=g(t)\,,
  \]
  show that $z(t)=\alpha y_f(t)+\beta y_g(t)$ is a solution of
  \[
  y''+a(t)y'+b(t)y=\alpha f(t)+ \beta g(t)\,,
  \]
  where $\alpha$ and $\beta$ are any real numbers, by plugging it into
  the ODE.


% Polking 4.3 #38
\item As we've seen, to solve ODE of the form
  \[
  y''+by'+cy=0\,,\qquad\mbox{$b$ and $c$ constants}
  \]
  we assume that the solution has the form $e^{rt}$, and then we plug
  this back into the ODE to get the \emph{characteristic equation}:
  $r^2+br+c=0$.  Given that this equation has a double root $r=r_1$
  (i.e., the roots are $r_1=r_2$), show by direct substitution that
  $y=te^{rt}$ is a solution of the ODE, and then write down the
  general solution.


\item Suppose a glass of water sitting in a room of constant
  temperature $A$ obeys Newton's law of cooling. We can express this
  by the simple differential equation $T'=k(A-T)$, where $T(t)$ is
  the temperature of the water at time $t$.
  \begin{enumerate}
    \item Solve the homogeneous equation, $T_h'=-kT_h$. What is the
      physical interpretation of $T_h(t)$?
    \item Find a particular solution $T_p(t)$ to the original equation
      using your physical intuition alone. 
    \item Find the general solution by adding the solutions you got in
      Parts (a) and (b). 
  \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}


\item Suppose we have a tank containing $150$ gallons of solution with
  an unknown salt concentration. Suppose water with a salt
  concentration of $2$ oz/gal flows in at a rate of $3$ gal/min, and
  water (fully mixed) drains from the tank at the same rate.  We wish
  to determine the salt content $x(t)$ at any point in time. Recall
  that we can do this by writing an ODE:
  \begin{align*}
    x'(t)&=(\mbox{rate in})-(\mbox{rate out}) \\
    &=(\mbox{flow rate in})(\mbox{incoming concentration})-
    (\mbox{flow rate out})(\mbox{outgoing concentration}) \\ 
    &=(3\mbox{ gal/min})(2\mbox{ oz/gal})-(3\mbox{
      gal/min})\frac{x(t)\mbox{ oz}}{150\mbox{ gal}} \\
    &=6-\frac{1}{50}x(t)\,.
 \end{align*}
  All that is left is solving the ODE $x'=6-\frac{1}{50}x$ for $x(t)$.
  \begin{enumerate}
  \item Sketch this physical situation.
  \item Solve the related homogeneous equation, $x_h'=-\frac{1}{50}x_h$.
  \item Find a simple particular solution $x_p(t)$ to the original
    ODE. Describe the physical situation that this represents.
  \item Find the general solution of this differential equation. 
  \end{enumerate}
  
\end{enumerate}

\end{document}
