\magnification=\magstep1
\input epsf
\def\everymath{\displaystyle}
\def\problemspace{0.5truein}

\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\X{{\bf X}}
\def\Y{{\bf Y}}
\def\U{{\bf U}}
\def\A{{\bf A}}
\def\G{{\bf G}}
\def\I{{\bf I}}
\def\P{{\bf P}}
\def\0{{\bf 0}}


%% TITLE
\noindent
{\bf Math 325: Differential Equations}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl December 19, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}

 0.5truein\noindent
Record your answers by placing an $\times$ through one letter for each
problem on this answer sheet.  There are 20 multiple choice questions
worth 6 points each. You start with 30~points.

\def\answerspace{0.3in}

%% PROBLEM
Find the general solution of $y^{(4)} + y^{(2)} = 0$.

%% CORRECT
$y = c_1 + c_2 t + c_3 \cos(t) + c_4 \sin(t)$
%% WRONG
$y = c_1t + c_2 t^2 + c_3 \cos(t) + c_4 \sin(t)$
%% WRONG
$y = (c_1 + c_2 t)\cos(t) + (c_3 + c_4t)\sin(t)$
%% WRONG
$y = (c_1t + c_2 t^2)\cos(t) + (c_3t + c_4t^2)\sin(t)$
%% WRONG
$y = c_1t + c_2 t^2 + c_3 e^t + c_4 e^{-t}$

%% PROBLEM
Determine the form  of a particular solution to the non-homogeneous
equation
$$y''' - 4y'' + 4y' = t + e^{2t}$$

%% CORRECT
$y = At^2 + Bt + C t^2 e^{2t}$
%% WRONG
$y = At^3 + Bt^2 + C t e^{2t}$
%% WRONG
$y = At^2 + Bt + C t e^{2t}$
%% WRONG
$y = At^3 + Bt^2 + C t^2 e^{2t}$
%% WRONG
$y = At^3 + Bt^2 + (C t^2  + Dt)e^{2t}$

%% PROBLEM
Given that $\X = c_1 \bmatrix{2\cr-1} + c_2\bmatrix{1\cr2}e^{5t}$ is
general solution of the homogeneous system $\X' = \A\cdot\X$, use the
method of variation of parameters to find a particular solution of the
non-homogeneous system $\X' = \A\cdot\X + \bmatrix{2 \cr -1}e^t$.

%% CORRECT
$\X = \bmatrix{2\cr-1}e^t$
%% WRONG
$\X = \bmatrix{2\cr-1}$
%% WRONG
$\X = \bmatrix{1\cr2}e^{t}$
%% WRONG
$\X = \bmatrix{2e^{5t}\cr-e^t}$
%% WRONG
$\X = \bmatrix{e^t\cr2e^{5t}}$

%% PROBLEM
Find the fundamental matrix $\Phi(t)$ with $\Phi(0) = \I$ for the system
$$\X' = \bmatrix{3 & 6\cr -1 & -2}\cdot\X$$

%% CORRECT
$\bmatrix{3e^t-2 & 6e^t-6\cr 1 - e^t & 3 - 2e^t}$
%% WRONG
$\bmatrix{1 & 3-3e^t\cr t & e^t}$
%% WRONG
$\bmatrix{\cos(t) & \sin(2t)\cr  -\sin(t) & \cos(2t)}$
%% WRONG
$\bmatrix{3e^t-2 & 6e^t-6\cr e^t-1 & e^t}$
%% WRONG
$\bmatrix{\cos(3t)& \sin(t)\cr \sin(3t) & \cos(t)}$

%% PROBLEM
Determine which of the following vectors is an eigenvector of the matrix
$$\bmatrix{0 & 0 & -1\cr2 & 0 & 0\cr -1 & 2 & 4\cr}$$
corresponding to the eigenvalue $r=1$.

%% CORRECT
$\bmatrix{1 \cr 2 \cr -1}$
%% WRONG
$\bmatrix{2 \cr 1 \cr -1}$
%% WRONG
$\bmatrix{1 \cr 0 \cr -1}$
%% WRONG
$\bmatrix{1 \cr -1 \cr 2}$
%% WRONG
$\bmatrix{1 \cr -1 \cr 0}$

%% PROBLEM
Solve the initial value problem
$$\X' = \bmatrix{0 & 2\cr -2 & 0}\cdot\X,\quad \X(0) = \bmatrix{3\cr 4}$$

%% CORRECT
$\bmatrix{3\cos(2t) + 4\sin(2t)\cr -3\sin(2t) + 4\cos(2t)}$
%% WRONG
$\bmatrix{3\cos(2t) \cr 4\cos(2t)}$
%% WRONG
$\bmatrix{3\cr 4}e^{2t}$
%% WRONG
$\bmatrix{3\cosh(2t) \cr 4\cosh(2t)}$
%% WRONG
$\bmatrix{4\cos(2t) - 3\sin(2t)\cr 3\cos(2t) + 4\sin(2t)}$

%% PROBLEM
Find the critical points of the system
$$\eqalign{
{dx\over dt} &= x^2 + xy\cr
{dy\over dt} &= y^3 + x^2\cr}$$

%% CORRECT
$(0,0)$, $(1,-1)$
%% WRONG
$(0,0)$, $(1,-1)$, $(-1,1)$
%% WRONG
$(0,0)$, $(1,1)$, $(-1,-1)$
%% WRONG
$(0,0)$, $(1,-1)$, $(1,1)$, $(-1,1)$, $(-1,-1)$
%% WRONG
$(0,0)$, $(-1,1)$

%% PROBLEM
Determine the type and stability of the critical point $(1,1)$ for the
system
$$\eqalign{
{dx\over dt} &= 2x - 3x^2y   +  x^3\cr
{dy\over dt} &=  y +  x  y^2 - 2y^3\cr}$$

%% CORRECT
stable spiral
%% WRONG
stable node
%% WRONG
unstable node
%% WRONG
unstable saddle point
%% WRONG
unstable spiral

%% PROBLEM
The following nonlinear system models populations of competing species:
  $$\eqalign{
  {dx\over dt} &= x(2 -  y - x)\cr
  {dy\over dt} &= y(3 - 2x - y)}$$
Determine which phase portrait corresponds to this system.

%% CORRECT
\vtop{\hsize = 2.1truein\epsfxsize=2in\epsfbox{vf5.eps}}
%% WRONG
\vtop{\hsize = 2.1truein\epsfxsize=2in\epsfbox{vf1.eps}}
%% WRONG
\vtop{\hsize = 2.1truein\epsfxsize=2in\epsfbox{vf2.eps}}
%% WRONG
\vtop{\hsize = 2.1truein\epsfxsize=2in\epsfbox{vf3.eps}}
%% WRONG
\vtop{\hsize = 2.1truein\epsfxsize=2in\epsfbox{vf4.eps}}

%% PROBLEM
Find the Fourier series for the function
$f(x) = \cases{1 - x & $0 \le x \le 1$\cr
-1-x & $-1 \le x \le 0$}$
$$\epsfxsize=2truein\epsfbox{plot.eps}$$

%% CORRECT
$f(x) = {2\over \pi}\sin( \pi x) + {2\over2\pi}\sin(2\pi x) + 
        {2\over3\pi}\sin(3\pi x) + {2\over4\pi}\sin(4\pi x) +  \cdots$
%% WRONG
$f(x) = {3\over \pi}\sin( \pi x) + {1\over2\pi}\sin(2\pi x) + 
        {3\over3\pi}\sin(3\pi x) + {1\over4\pi}\sin(4\pi x) +  \cdots$
%% WRONG
$f(x) = {2\over \pi}\sin( \pi x) - {1\over2\pi}\sin(2\pi x) + 
        {2\over3\pi}\sin(3\pi x) - {1\over4\pi}\sin(4\pi x) +  \cdots$
%% WRONG
$f(x) = {2\over \pi}\sin( \pi x) - {2\over3\pi}\sin(3\pi x) + 
        {2\over5\pi}\sin(5\pi x) - {2\over7\pi}\sin(7\pi x) +  \cdots$
%% WRONG
$f(x) = {1\over \pi}\sin( \pi x) + {2\over3\pi}\sin(3\pi x) + 
        {1\over5\pi}\sin(5\pi x) + {2\over7\pi}\sin(7\pi x) +  \cdots$

%% PROBLEM
Use the method of separation of variables to replace the partial
differential equation $$(x u_x)_x + u_{tt}= 0$$ by a pair of ordinary
differential equations.

%% CORRECT
$\left\{\eqalign{&x X''(x) + X'(x) - \sigma X(x) = 0\cr
&T''(t) + \sigma T(t) = 0}\right.$
%% WRONG
$\left\{\eqalign{&X''(x) - \sigma X(x) = 0\cr
&T''(t) - \sigma T(t) = 0}\right.$
%% WRONG
$\left\{\eqalign{&X''(x) + x X'(x) - \sigma X(x) = 0\cr
&T''(t) + \sigma = 0}\right.$
%% WRONG
$\left\{\eqalign{&X''(x) - \sigma x X'(x) = 0\cr
&T''(t) + \sigma  t T(t) = 0}\right.$
%% WRONG
$\left\{\eqalign{&X''(x) + \sigma x X'(x) = 0\cr
&T''(t) - \sigma  T(t) = 0}\right.$

%% PROBLEM
Solve the heat equation $5u_{xx} = u_{t}$ for $0\le x \le 10$, $t \ge 0$,
subject to the boundary conditions $u(0,t) = 0$, $u(10,t) =
0$ for $t\ge 0$ and the initial condition $u(x,0) = 30$ for $0 \le x
\le 10$.

%% CORRECT
$u(x,t) = {120\over\pi} \sum_{k=1}^\infty {1\over 2k-1}
\sin({(2k-1)\pi\over10}x) e^{-((2k-1)\pi/10)^2 5t}$
%% WRONG
$u(x,t) = {60\over \pi} \sum_{k=1}^\infty {1\over 2k}
\sin({2k\pi\over10}x) e^{-(2k\pi/10)^2 5t}$
%% WRONG
$u(x,t) = {120\over \pi} \sum_{k=1}^\infty {1\over k}
\sin({k\pi\over10}x) e^{-(k\pi/10)^2 \sqrt{5}t}$
%% WRONG
$u(x,t) = {60\over \pi} \sum_{k=1}^\infty {(-1)^k\over k}
\sin({k\pi\over10}x) e^{-(k\pi/10)^2 \sqrt{5}t}$
%% WRONG
$u(x,t) = {120\over \pi} \sum_{k=1}^\infty {(-1)^k\over 2k-1}
\sin({k\pi\over10}x) e^{-(k\pi/10)^2 t}$

%% PROBLEM
Solve the wave equation $9u_{xx} = u_{tt}$ for $0 \le x \le \pi$, $t \ge
0$, subject to the boundary conditions $u(0,t) = 0$, $u(\pi,t) = 0$ for
$t \ge 0$ and the initial conditions $u(x,0) = -x$, $u_t(x,0) = 0$ for
$0 \le x \le \pi$.

%% CORRECT
$u(x,t)=2\sum_{n=1}^\infty {(-1)^n \over n} \sin(nx)\cos(3nt)$
%% WRONG
$u(x,t)=2\sum_{n=1}^\infty {(-1)^n \over n} \sin(nx)\cos(nt)$
%% WRONG
$u(x,t)=\sum_{n=1}^\infty {(-1)^n \over n\pi} \sin({n\pi\over3}x)
\cos({n\over3}t)$
%% WRONG
$u(x,t)=\sum_{n=1}^\infty {1 \over n\pi} \sin({n\pi\over3}x)\cos({n\pi\over3}t)$
%% WRONG
$u(x,t)={2\over3\pi}\sum_{n=1}^\infty {1 \over n^2} \sin(nx)\cos(3nt)$

%% PROBLEM
Solve the heat equation $u_{xx} = u_{t}$, $0\le x \le 3$, $t \ge 0$,
subject to the boundary conditions $u(0,t) = 0$, $u(3,t) = 3$ for $t\ge
0$ and the initial condition $u(x,0) = x$ for $0 \le x \le 3$.

%% CORRECT
$u(x,t)=x$
%% WRONG
$u(x,t)=x + {6\over \pi}\sum_{n=1}^{\infty}{(-1)^n\over n}\sin({n\pi\over3}x)
e^{-(n\pi/3)^2t}$
%% WRONG
$u(x,t)=-{6\over \pi}\sum_{n=1}^{\infty}{(-1)^n\over n}\sin({n\pi\over3}x)
e^{-(n\pi/3)^2t}$
%% WRONG
$u(x,t)={6\over \pi}\sum_{n=1}^{\infty}{(-1)^n\over n}\sin({n\pi\over3}x)
\cos({n\pi\over3}t)$
%% WRONG
$u(x,t)={2\over \pi}\sum_{n=1}^{\infty}{(-1)^n\over n}\sin({n\pi\over3}x)
\sin({n\pi\over3}t)$

%% PROBLEM
Find the inverse Laplace transform of the function
${4\over(s-1)^3}$.

%% CORRECT
$2t^2e^t$
%% WRONG
$2t^2$
%% WRONG
$4t^2e^{4t}$
%% WRONG
$te^t$
%% WRONG
$t^3e^t$

%% PROBLEM
Find the inverse Laplace transform of ${1-e^{-2s}\over s^2}$.

%% CORRECT
$t - u_2(t)(t-2)$
%% WRONG
$t^2 - u_2(t)(t-2)$
%% WRONG
$t - u_2(t)t^2$
%% WRONG
$t - u_2(t)t$
%% WRONG
$t^2 - u_2(t)\sin(t-2)$

%% PROBLEM
Determine the solution of the initial value problem
$$y'' + 2y' + 2y = \delta(t - \pi),\quad y(0) = 1,\quad y'(0) =  0$$

%% CORRECT
$y(t) = e^{-t}(\cos(t) + \sin(t)) + u_\pi(t)e^{-t+\pi}\sin(t-\pi)$
%% WRONG
$y(t) = e^{-2t}(\cos(2t) + \sin(2t)) +u_\pi(t)e^{-t+\pi}\sin(t-\pi)$
%% WRONG
$y(t) = 2u_\pi(t)e^{-t+\pi}\sin(t)$
%% WRONG
$y(t) = e^{-t}(\cos(t) + \sin(t))$
%% WRONG
$y(t) = e^{-t}(\cos(t) + \sin(t)) + u_\pi(t)\sin(t-\pi)$

%% PROBLEM
Find the general solution of the system
$$\X' = \bmatrix{4 & -3 \cr 8 & -6}\cdot\X$$

%% CORRECT
$\X = c_1\bmatrix{3\cr 4} + c_2\bmatrix{1\cr 2}e^{-2t}$
%% WRONG
$\X = c_1\bmatrix{3\cr 4}te^{-2t} + c_2\bmatrix{1\cr 2}e^{-2t}$
%% WRONG
$\X = c_1\bmatrix{4\cr -3}e^{-2t} + c_2\bmatrix{2\cr -1}e^{-2t}$
%% WRONG
$\X = c_1\bmatrix{3\cr 4}e^{t} + c_2\bmatrix{1\cr 2}e^{-t}$
%% WRONG
$\X = c_1\bmatrix{4\cr -3} + c_2\bmatrix{2\cr -1}e^{-2t}$

%% PROBLEM
Compute the Wronskian of
$$\X_1 = \bmatrix{\cos(t)\cr \sin(t)}e^t,\quad
\X_2 =\bmatrix{\sin(t)\cr \cos(t)}e^t$$

%% CORRECT
$e^{2t}(\cos^2(t) - \sin^2(t))$
%% WRONG
$e^{t}(\cos^2(t) - \sin^2(t))$
%% WRONG
$e^{2t}$
%% WRONG
$e^t$
%% WRONG
0

%% PROBLEM
Use the Euler method with step size $h = 0.5$ to find an approximation
of $\phi(1)$ where $\phi(t)$ is the solution to the initial value problem
$$y' = t^2 + y^2,\quad y(0) = 1$$

%% CORRECT
2.75
%% WRONG
1.25
%% WRONG
1.50
%% WRONG
4.0
%% WRONG
2.25

%% END
\end
