% 2018 Differential Equations 2280-2
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\begin{center}\Large\bf
Differential Equations 2280 \\
\ifsolutions
\small Midterm Exam 2 with Solutions\\
\else
\small Midterm Exam 2 Review: Problems Only\\
\fi
\small Exam Date: 30 March 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.

\begin{exercises}
\prob{1.}{(Chapter 3)} ~\newline
\BF{(a)} [60\%] Find by any applicable method
the steady-state periodic solution. 

\BF{(b)} [40\%] Variation of parameters: electrical/mechanical system.
\begin{quote}
{\bf Expected in (b)}: {\small\sf (1) Characteristic equation and Euler atoms. (2) Wronskian, (3) Explicit formula for $y_h$. (4) Integral formula for $y_p$, simplified as much as possible, integrals unevaluated.}
\end{quote}

\ifsolutions
\Answer{~\newline
}
\fi

\end{exercises}
\EJECT

\begin{exercises}
\prob{2.}{(Laplace Theory)} ~\newline
\BF{(a)} [50\%] Assume $f(t)$ is of exponential order. Find $f(t)$ in the given relation.

\bigskip

\BF{(b)} [50\%] Solve by Laplace's method: second order forced spring-mass system with initial conditions.

\ifsolutions
\Answer{~\newline
\BF{(a)}  \\
\BF{(b)}
}
\fi
\end{exercises}
\EJECT
\begin{exercises}
\prob{3.}{(Laplace Theory)} ~\newline
   \BF{(a)} [40\%] Solve a linear third order homogeneous differential equation by
 Laplace's Method.
 \begin{quote}
 {\bf Expected in (a)}: {\small\sf (1) Laplace's method steps to find $\L(x(t))$, (2) Backward table steps to find $x(t)$, (3) An answer check.}
 \end{quote}


  \BF{(b)} [60\%] Solve a $2\times 2$ system of non-homogeneous linear differential equations
 by Laplace's Method.

\begin{quote}
 {\bf Expected in (b)}: {\small\sf (1) Laplace's method steps to find a system of linear algebraic equations for $\L(x(t))$ and $\L(y(t))$. (2) Cramer's Rule steps to find $\L(x(t))$ and $\L(y(t))$. (3) Backward Laplace table steps to find $x(t),y(t)$. (4) Check the answer for sanity, e.g., compute the Euler atoms.}
 \end{quote}

\ifsolutions
\Answer{~\newline
(a)
\\
(b)
}
\fi
\end{exercises}
\EJECT

\begin{exercises}
\prob{4.}{(Systems of Differential Equations)} ~\newline
 The Eigenanalysis Method (section 5.2) says that, for a $3\times 3$ system
$\dd\frac{d}{dt}\vec{u}(t) = A\vec{u}(t)$, the general solution is $\vec{u}(t)=c_1\.v_1e^{\lambda_1
t}+c_2\.v_2e^{\lambda_2 t}+c_3\.v_3e^{\lambda_3 t}$. In the
solution formula, $(\lambda_1,\.v_1)$,  $(\lambda_2,\.v_2)$, $(\lambda_3,\.v_3)$ are eigenpairs
of $A$.  Assume given the two $3\times 3$ matrices
$$A=\left[\begin{array}{rrr}
 * & * &  *  \\[0pc]
 * & * &  *  \\[0pc]
 * & * &  *
 \end{array}\right],\quad\quad
 B=\left[\begin{array}{rrr}
 * & * &  *  \\[0pc]
 * & * &  *  \\[0pc]
 * & * &  *
 \end{array}\right].$$

 \BF{(a)} [50\%] Matrix $A$ has only two eigenpairs. Display eigenanalysis details for $A$. Then explain method failures
 in methods (1) to (4).

\bigskip

 \BF{(b)} [50\%] Find by any method the general solution of $\dd\frac{d}{dt}\vec{u}(t) = B\vec{u}(t)$.

\ifsolutions
\Answer{~\newline
(a): \\
(b):
}
\fi
\end{exercises}
\EJECT

\begin{exercises}
\prob{5.}{(Systems of Differential Equations)}
\newline
A system $\frac{d}{dt}\vec{u} = A\vec{u}$ with $A$ an $n\times n$ real matrix can be solved by many methods. Consider below {\bf only these possibilities}:
\begin{quote}
{\small\sf (1) Cayley-Hamilton-Ziebur method, from section 4.2. (2) Eigenanalysis method from section 5.2. (3) Laplace's method, from chapter 7. (4) First order linear integrating factor method from section 1.5.}
\end{quote}

\BF{(a)} [20\%] Given an example system, which of methods  (1), (3), (4) might be possible to use? Don't do any details or illustrations: kindly write a sentence and explain your logic.

\bigskip

\BF{(b)} [40\%] Solve for the general solution by the Cayley-Hamilton-Ziebur shortcut (chapters 4 and 5):
$$\dd\frac{d}{dt}\vec{u}=B\vec{u}, \quad\quad
B=\left[\begin{array}{rrrr}
 * & *   \\[0pc]
 * & *
 \end{array}\right]\quad\quad\mbox{\bf (eigenvalues = $*$, $*$)}.
 $$

\bigskip


\BF{(c)} [40\%] Choose any method applicable and solve for the general solution of system
$$
\left\{
\begin{array}{lcl}
x'(t) &=& *x(t) + *y(t) + *z(t), \\
y'(t) &=& *x(t) + *y(t) + *z(t), \\
z'(t) &=& *x(t) + *y(t) + *z(t).
\end{array}
\right.
$$


\Answer{~\newline
(a) \\
(b)\\
(c)
}
\end{exercises}
\EJECTEND

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