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\begin{center}\Large\bf
Differential Equations 2280 \\
\small Midterm Exam 3\\
\small Exam Date: 13 April 2018 at 12:50pm
\end{center}


{\bf Instructions}: This in-class exam is 50 minutes. No
calculators, notes, tables or books. No answer check is expected.
Details count 3/4, answers count 1/4.
\subsection*{Chapter 3 -- $N$th Order Differential Equations}

\BF{Problem (1a)} [40\%]  Find the {\bf Beats} solution for
a forced undamped spring-mass problem
$$ x'' + \sigma^2x = F_0\cos(\omega t),\quad x(0)=x'(0)=0.$$
% de:=diff(x(t),t,t)+256*x(t)=247*cos(3*t), x(0)=0,D(x)(0)=0;
It is known that this solution is the sum of two harmonic oscillations of different frequencies.
{\bf To save time}, please don't convert your answer.
\nextpage
%\subsection*{Chapter 3 -- $N$th Order Differential Equations}

\bigskip

 \BF{Problem (1b)} [30\%] Let $f(x)$ be a given linear combination of Euler solution atoms. Find the characteristic equation of a linear
 homogeneous scalar differential equation of least order such that $y=f(x)$ is a solution. Kindly leave the characteristic
 equation in factored form, unexpanded.

\nextpage
%\subsection*{Chapter 3 -- $N$th Order Differential Equations}

\bigskip

\BF{Problem (1c)} [40\%]
Consider a forced mechanical oscillation equation and/or a forced electrical current equation.
Determine the practical resonance frequency $\omega$ for each equation. Determine a particular solution
by the method of undetermined coefficients. Find the amplitude of this particular solution.

\ifsolutions
\Answer{%2018
}
\fi
\EJECT

\subsection*{Chapters 4 and 5 -- Systems of Differential Equations}

\begin{quote}\small\sf
\BF{Theorem. (Eigenanalysis Method)} If $A$ is a real $3\times 3$ matrix with eigenpairs
 $(\lambda_1,\.v_1)$,  $(\lambda_2,\.v_2)$,  $(\lambda_3,\.v_3)$, then the system
$\.x' = A\.x$ has general solution
$$\.x(t)=c_1\.v_1e^{\lambda_1
t}+c_2\.v_2e^{\lambda_2 t}+c_3\.v_3e^{\lambda_3 t}.$$
\BF{Theorem. (Cayley-Hamilton-Ziebur)}.
The components of solution $\.x$ of $\.x'(t)=A\.x(t)$ are linear combinations of Euler solution atoms obtained from the roots of the characteristic equation $|A-\lambda I|=0$.

\BF{Definition}. Let $A$ be an $n\times n$ real matrix. An augmented matrix $\Phi(t)$ of $n$ independent solutions of $\.x'(t)=A\.x(t)$ is called a
 \BF{fundamental matrix}. It is known that the general solution is $\.x(t)=\Phi(t)\.c$, where $\.c$ is a column vector of arbitrary constants $c_1, \ldots, c_n$. An alternate and widely used definition of fundamental matrix is  $\Phi'(t) = A\Phi(t)$, $|\Phi(0)| \ne 0$.
\end{quote}
\eject
\subsection*{Chapters 4 and 5 -- Systems of Differential Equations}
 \BF{Problem (2a)} [30\%]  Assume given a specific $3\times 3$ matrix $A$
with given eigenvalues $\lambda_1,\lambda_2,\lambda_3$.
Apply the Cayley-Hamilton-Ziebur theorem to this example.
\nextpage
%\subsection*{Chapters 4 and 5 -- Systems of Differential Equations}

\bigskip

\BF{Problem (2b)} [40\%]   A linear cascade, typically found in brine tank models, satisfies  $\frac{d}{dt}\vec{x}(t)=A\vec{x}(t)$ where the $4\times 4$ matrix and vector $\.x$ are defined by
$$A=\left(\begin{array}{rrrr}
 * & * &  *  & * \\{}
 * & * &  *  & * \\{}
 * & * &  *  & * \\{}
 * & * &  *  & *
\end{array}\right),\quad \.x = \colvector{x_1\\ x_2\\ x_3\\ x_4}.$$
Use an appropriate method to find the vector general solution $\vec{x}(t)$ of $\frac{d}{dt}\vec{x}(t)=A\vec{x}(t)$.


\nextpage

\bigskip

%\subsection*{Chapters 4 and 5 -- Systems of Differential Equations}
\BF{Problem (2c)} [40\%]  A linear cascade, typically found in brine tank models, satisfies  $\frac{d}{dt}\vec{x}(t)=A\vec{x}(t)$ where the $4\times 4$ matrix and vector $\.x$ are defined by
$$A=\left(\begin{array}{rrrr}
 * & * &  *  & * \\{}
 * & * &  *  & * \\{}
 * & * &  *  & * \\{}
 * & * &  *  & *
\end{array}\right),\quad \.x = \colvector{x_1\\ x_2\\ x_3\\ x_4}, \quad \.x(0)=\colvector{1\\ 0\\ 0\\ 0}.$$
Apply Laplace's method to obtain a $4\times 4$ system for $\L(x_1), \L(x_2),\L(x_3),\L(x_4)$.  Your solution can use scalar equations or the vector-matrix equation $\frac{d}{dt}\vec{x}(t)=A\vec{x}(t)$. Other parts of this problem: Solve the system using Cramer's Rule.  Solve for $\.x$ using Laplace tables and Lerch's theorem.

\nextpage

\bigskip

%\subsection*{Chapters 4 and 5 -- Systems of Differential Equations}

   \BF{Problem (2d)} [30\%] The Cayley-Hamilton-Ziebur shortcut is to be applied to a system
   $$ x' = ax + by,\quad y' = cx + dy,$$
   where $a,b,c,d$ are given along with the eigenvalues $\lambda_1,\lambda_2$.

\begin{quote}\sf
   \BF{Part 1}. Show the details of the method, finally displaying formulas for $x(t), y(t)$.

    \BF{Part 2}. Report a fundamental matrix $\Phi(t)$.

    \BF{Part 3}. Use \BF{Part 2} to find the exponential matrix $e^{At}$.
 \end{quote}
\ifsolutions
\Answer{%2018
}
\fi

\nextpage
\eject
\subsection*{Chapter 6, Linear and Nonlinear Dynamical Systems}

\BF{Problem (3a)} [20\%]
 Determine whether the unique equilibrium $\vec{u}=\vec{0}$ is stable or
unstable. Then classify the equilibrium point $\vec{u}=\vec{0}$ as a
saddle, center, spiral or node. Sub-classification into improper or proper node is not required.
 $$\frac{d}{dt}\vec{u}=\twobytwo{*&*}{*&*}\vec{u}$$
\nextpage
%\subsection*{Chapter 6, Linear and Nonlinear Dynamical Systems}

\bigskip

\BF{Problem (3b)} [30\%] Consider the nonlinear dynamical system
  $$\begin{array}{lcl}
  x'&=& *, \\
   y'&=& *.
  \end{array}
 $$
 An equilibrium point is $x=*$, $y=*$. Compute
 the Jacobian matrix of the linearized system at this
 equilibrium point.
% f:=x-2*y*y-2*y+32;g:=2*x*(x-2*y); solve([f=0,g=0],[x,y]); [-8,4], [8,-4]
\nextpage
%\subsection*{Chapter 6, Linear and Nonlinear Dynamical Systems}

\bigskip

\BF{Problem (3c)} [30\%] Consider the nonlinear system
  $\dd\left\{\begin{array}{lcl}
  x'&=& *, \\
   y'&=& *.
  \end{array}\right.$\\
 %At equilibrium point $x=0$, $y=0$, the Jacobian matrix is
 %$A=J(0,0)=\left.\twobytwo{0&1}{-5+\frac{15}{4}x^3&-2}\right|_{x=0}=\twobytwo{0&1}{-5+\frac{15}{4}x^2&-2}$.
 %$\dd A=J(0,0)=\twobytwo{0&1}{-5&-2}$.
 \begin{quote}
   {\bf (Part 1)}~~
  Determine the stability at $t=\infty$ and the phase portrait
  classification saddle, center, spiral
  or node at $\vec{u}=\vec{0}$ for the {\bf linear dynamical system} $\frac{d}{dt}\vec{u}=A\vec{u}$, where $A$ is the Jacobian matrix of this system at $x=*$, $y=*$.

   {\bf (Part 2)}~~
  Apply the Pasting Theorem to classify $x=2$, $y=0$ as a saddle, center, spiral
  or node for the {\bf nonlinear dynamical system}. Discuss all details of the application of the theorem.
  {\em Details count 75\%}.

  % {\bf (3)}~~ Repeat the classification details of the previous two parts (1), (2) for the other two equilibrium points $(2,0), (-2,0)$, for which the Jacobian is
%$A=J(\pm2,0) = \twobytwo{0&1}{10&-2}$.
  \end{quote}

\nextpage
%\subsection*{Chapter 6, Linear and Nonlinear Dynamical Systems}

\BF{Problem (3d)} [20\%]  State the hypotheses and the conclusions of the {\em Pasting Theorem} used in problem {\bf (3c)} above. Accuracy and completeness expected.

\bigskip
\ifsolutions
\Answer{%2018
}
\fi
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