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Midterm 3 2280 8:35}}}
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\begin{center}\Large\bf
Differential Equations 2280\\
Midterm Exam 3\\
Wednesday, 22 April 2009
\end{center}

{\bf Instructions}: This in-class exam is 50 minutes. No
calculators, notes, tables or books. No answer check is expected.
Details count $75$\%. The answer counts $25$\%. Each problem is
scored 100.

Please discard this sheet after reading it.

\EJECT
\begin{exercises}
\prob{1.}{(ch7)} Arrowsmith [Page]~\newline
 (a) [25\%] Display the integral formula for the direct Laplace
 Transform of $\dd\frac{t}{1+t}$. Explain why the integral
 exists, citing theorems.
 \newline
 (b) [25\%] Derive the formula $\L(f'(t))=s\L(f(t))-f(0)$. 
 \newline
 (c) [50\%] Solve $x'''+3x''+2x'=0$, $x(0)=x'(0)=0$, $x''(0)=1$ by
 Laplace's Method.
  \newline
 (d) [50\%] Solve the system $x'=x+y$, $y'=x$, $x(0)=1$, $y(0)=0$
 by Laplace's Method.

\end{exercises}

\EJECT

\begin{exercises}
\prob{2.}{(ch5)} Cummings [Beattie]~\newline
 The eigenanalysis method says that the system
$\.x' = A\.x$ has general solution $\.x(t)=c_1\.v_1e^{\lambda_1
t}+c_2\.v_2e^{\lambda_2 t}+c_3\.v_3e^{\lambda_3 t}$. In the
solution formula, $(\lambda_i,\.v_i)$, $i=1,2,3$, is an eigenpair
of $A$.  Given
$$A=\left[\begin{array}{rrr}
 5 & 1 &  1  \\
 1 & 5 &  1  \\
 0 & 0 &  7
 \end{array}\right],$$
 then \newline
 (a) [75\%] Display eigenanalysis details for $A$. \newline
 (b) [25\%] Display the solution $\.x(t)$ of $\.x'(t)=A\.x(t)$.


\end{exercises}
\EJECT

\begin{exercises}
\prob{3.}{(ch5)} Karrens [Harris]~\newline
(a) [50\%]  The eigenvalues are $2$, $3$, $4$, $5$ for the matrix
$A=\left[\begin{array}{rrrr}
 4 & 1 & -1 & 0 \\
 1 & 4 & -2 & 1 \\
 0 & 0 &  2 & 0 \\
 0 & 0 &  2 & 4
 \end{array}\right]$.
%with(linalg):
%B:=matrix([[4,1,-1,0],[1,4,-2,1],[0,0,2,0],[0,0,2,4]]);
%eigenvals(B);
%dsolve({diff(y(x),x)=2*y(x),y(0)=1},y(x));
%dsolve({diff(y(x),x)=3*y(x)+exp(2*x),y(0)=0},y(x));
%dsolve({diff(y(x),x)=4*y(x)+(-exp(-x)+1)*exp(3*x),y(0)=0},y(x));
%dsolve({diff(y(x),x)=5*y(x)+(1/2*exp(-2*x)-exp(-x)+1/2)*exp(4*x),y(0)=0},y(x));

Display the general solution of $\.u'=A\.u$ according to Putzer's
spectral formula. Don't expand matrix products, in order to save time.
However, do compute the coefficient functions $r_1$, $r_2$, $r_3$. Given below is the 
answer for $r_4$, to shorten the computation.

$$ r_4(t) = -\frac16 e^{2t}+\frac12 e^{3t}-\frac12 e^{4t}+\frac16 e^{5t}.$$

(b) [50\%] Using the same matrix $A$ from part (a), display the solution of $\.u'=A\.u$ according to the Cayley-Hamilton Method. To save time, write out the system to be solved for the four vectors, and then stop, without solving for the vectors.


\ifsolutions
\begin{solution}{3} ~\newline
(a) Subtract $\lambda$ from the diagonal elements of $A$ and expand
the determinant $\det(A-\lambda I)$ to obtain the characteristic
polynomial $(1-\lambda)(1-\lambda)(4-\lambda)(1-\lambda)=0$. The
eigenvalues are the roots: $\lambda=1,1,1,4$. Used here was the
{\em cofactor rule} for determinants. Sarrus' rule does not apply
for $4\times 4$ determinants (an error) and the triangular rule
likewise does not directly apply (another error).

% exp(t), t*exp(t), 1/2*t^2*exp(t),
% 1/27*exp(4*t)-1/27*exp(t)-1/9*t*exp(t)-1/6*t^2*exp(t)])

(b) Let
$$B=\left[\begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 1 & 1 & 0 & 0 \\
 0 & 1 &  1 & 0 \\
 0 & 0 &  1 & 4
 \end{array}\right].$$
Define functions $r_1$, $r_2$, $r_3$, $r_4$ to be the components of the
vector solution $\.r(t)$ to the initial value problem
$$\.r'=B\.r,\quad \.r(0)=\colvector{1\\ 0\\ 0\\ 0}.$$
Solving,
$$ r_1=e^t,\quad r_2=te^t, \quad r_3=t^2e^t/2,$$
$$ r_4=\frac{1}{27}e^{4t} - \frac{1}{27}e^t - \frac{1}{9}te^t - \frac{1}{6}t^2e^t.$$
Define
$$P_1=I,\quad P_2=A-I,\quad P_3=(A-I)^2, \quad P_4=(A-I)^3.$$
Then $ \.u=\left(r_1P1+r_2P_2+r_3P_3+r_4P_4\right)\.u_0$ implies
$$ \.u=\left(r_1I+r_2(A-I)+r_3(A-I)^2+r_4(A-I)^3\right)\.u_0$$
\end{solution}
\fi
\end{exercises}
\EJECT

\begin{exercises}
\prob{4.}{(ch5)} Wyatt [Williams]~\newline


(a) [50\%] Display the solution of
$\.u'=\twobytwo{3&4}{0&1}\.u$ according to the Laplace Resolvent Method. To save time, do not evaluate the constants in partial fractions.

(b) [50\%] Display the solution of
$\.u'=\twobytwo{3&4}{0&1}\.u$ according to the Eigenanalysis Method. 

(c) [50\%] Display the exponential matrix $e^{At}$ for the system
$\.u'=\twobytwo{3&4}{0&1}\.u$.

(d) [50\%] Display the variation of parameters formula for the system below, but do not do any integrations, in order to save time.
$$\.u'=\twobytwo{3&4}{0&1}\.u+\colvector{e^{-t}\\ 0}.$$

\end{exercises}
 \EJECT
\begin{exercises}
\prob{5.}{(ch6)} Bennett [Osborne]~\newline
 (a) [30\%] Define {\em stable equilibrium} for $\.u'=\.f(\.u)$, a nonlinear 2-dimensional
 system in which $\.f$ is continuously differentiable.
\newline
 (b) [40\%] Give an example of a linear 2-dimensional system with a stable spiral at equilibrium point $x=y=0$.
 Draw a representative phase diagram about $x=y=0$.
 \newline
 (c) [40\%] Give an example of a linear 2-dimensional system with a stable center at equilibrium point $x=y=0$.
 Draw a representative phase diagram about $x=y=0$.
 \newline
 (d) [40\%] Give an example of a linear 2-dimensional system with an unstable saddle at equilibrium point $x=y=0$.
 Draw a representative phase diagram about $x=y=0$.
 \newline
 (e) [30\%] Assume a $2$-dimensional predator-prey system $\.u'=\.f(\.u)$
 has equilibrium points $(0,0)$, $(160,0)$, $(0,180)$ and $(100,90)$.
 Explain the possible physical meanings of the equilibria, e.g., extinction,
 explosion, carrying capacity.


\end{exercises}
 \EJECTEND
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 \EJECT
\begin{exercises}
\prob{2.}{(ch6)} ~\newline
  (a) [40\%] Classify the linearization of the nonlinear system $x'=x+4y-xy^2$,
 $y'=2x-y+x^2y$ at equilibrium $x=y=0$ as node,
 spiral, center or saddle.
 \newline
 (b) [30\%] Is there a two-dimensional nonlinear system with a
 spiral at equilibrium $x=y=0$ while the linearization at $x=y=0$ is a
 center? Cite theorems or give an example.
 \newline
 (c) [30\%] If a two-dimensional system has a saddle at equilibrium
 $x=y=0$, then what are the possible classifications for the
 linearized system at $x=y=0$? Supply reasons and cite theorems.


\ifsolutions
\begin{solution}{2} ~\newline

\end{solution}
\fi
\end{exercises}
