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\begin{document}







\title
\noindent
{\bf Math 226: Calculus IV}\hfill
Sign your Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
%Put in the correct exam number and date%%%%%%%%
{\bf Exam II}\quad{\sl March 20, 1998}\hfill
Print your Name:\hbox to 2.5truein{\hrulefill}\medskip\hfill
Section:\hbox to 2.5truein{\hrulefill}\medskip\hfill
TA:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Do not turn this page until you are told to begin.
\bigskip\noindent
{\bf Do not detach the answer sheet from the test.}
\bigskip\noindent
You are required to hand in both the answer sheet and the problems.
\bigskip\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 16 multiple
choice questions worth 6 points each. You start with 10 points, and the
highest
possible score is 106.  Fill in the answers as you go along.  You will not be
allowed to fill in the answers after the time is up.
\bigskip\noindent
You may use a calculator, but only the standard functions found on very
inexpensive
scientific calculators.  In particular you may not use graphing,
integration, formula, matrix or
program capabilities.
\bigskip\noindent



\def\answerskip{\vskip0.25in}

\note %1
\problem
Let $y(t)$ be the solution of \quad $2y''+y'=0$ \quad for which \quad
$y(0)=y'(0)=1$.\quad  Find
$\lim\limits_{t\rightarrow\infty}y(t).$
\correct
3
\wrong
${3\over 2}$
\wrong
1
\wrong
4
\wrong
$\infty$



\note %2
\problem
The existence and uniqueness theorem for second order linear differential
equations guarantees a unique
solution to the equation
$$(t-1)^2y''+(\ln t)y'+3y=2$$
\noindent
satisfying
\correct
$y(2)=1, y'(2)=0$
\wrong
$y(0)=1, y'(0)=2$
\wrong
$y(1)=2, y'(1)=3$
\wrong
$y(3)=2, y(5)=4$
\wrong
$y(0)=0, y'(0)=0$


\note %3
\problem
Each of the following is a set of solutions of the equation \quad
$y''-9y=0$.\quad Find the one that is
${\bf NOT}$ a fundamental set of solutions.
\correct
$\{e^{3t-1},e^{3t+1}\}$
\wrong
$\{e^{3t},e^{-3t}\}$
\wrong
$\{e^{3t}+e^{-3t},e^{3t}-e^{-3t}\}$
\wrong
$\{e^{1-3t},e^{1+3t}\}$
\wrong
$\{3e^{3t},-e^{-3t}\}$

\note %4
\problem
Let $W(t)$ be the Wronskian of a fundamental set of solutions of the equation
$$(1+t^2)y''+ty'-(3t+2)y=0.$$
\noindent
If $W(0)=4$, find $W(\sqrt3)$.
\correct
2
\wrong
1
\wrong
8
\wrong
0.707
\wrong
$-4$



\note %5
\problem
Let $y(t)$ be the solution of the initial value problem
$$y''-2y'+17y=0,\quad y(0)=1=y'(0).$$
\noindent
Find $y({\pi\over 16})$.
\correct
0.861
\wrong
1.721
\wrong
-2.139
\wrong
1.217
\wrong
0


\note %6
\problem
Find the general solution of the equation \quad $4y''+12y'+9y=3e^{{3\over2}t}$.
\correct
$e^{-{3\over2}t}(c_1+c_2t)+{1\over12}e^{{3\over2}t}$
\wrong
$e^{-{3\over2}t}(c_1+c_2t)+{t\over6}e^{{3\over2}t}$
\wrong
$e^{-{3\over2}t}(c_1+c_2t+{t^2\over9})$
\wrong
$c_1e^{-{3\over2}t}+c_2t+{1\over 18}e^{{3\over2}t}$
\wrong
$e^{{2\over3}t}(c_1+c_2t)-{1\over 12}e^{{3\over2}t}$

\note %7
\problem
The function \quad $y_1(t)=e^t$\quad is a solution of the differential equation
$$ty''-y'-(t-1)y=0,\quad t>0.$$
\noindent
The function \quad $y_1(t)v(t)$\quad will be a second linearly independent
solution if $v(t)$ satisfies
the differential equation
\correct
$v''+(2-{1\over t})v'=0$.
\wrong
$tv''+(1-2t)v'=0$.
\wrong
$v''+{e^{2t}\over t}v'=0$.
\wrong
$(te^{-2t}v')'=0$.
\wrong
$v''+(1+{1\over t})v'=0$.



\note %8
\problem
Using the method of undetermined coefficients, find the correct form for a
particular solution $Y(t)$ of
the equation
$$y''+4y=e^{2t}\cos 2t-\sin 2t.$$
\correct
$e^{2t}(A\cos 2t+B\sin 2t)+t(C\cos 2t+D\sin 2t)$
\wrong
$e^{2t}(A\cos 2t+B\sin 2t)+C\cos 2t+D\sin 2t$
\wrong
$te^{2t}(A\cos 2t+B\sin 2t)$
\wrong
$(At+B)e^{2t}\cos 2t+(Ct+D)\sin 2t$
\wrong
$te^{2t}(A\cos 2t+B\sin 2t)+t(C\cos 2t+D\sin 2t)$


\note %9
\problem
Suppose that $y_1(x)=x$ and $y_2(x)=x^2$ are solutions of the differential
equation
$$y''+p(x)y'+q(x)y=0,$$
\noindent
where $p(x)$ and $q(x)$ are continuous for $x>0$.  Find a particular
solution to the nonhomogeneous
equation
$$y''+p(x)y'+q(x)y=10x^4.$$
\correct
${x^6\over 2}$
\wrong
${x^5\over 10}$
\wrong
${x^6\over 5}$
\wrong
${x^5\over 12}$
\wrong
${x^7\over 20}$

\note %10
\problem
An object which weighs 32 pounds is attached to a spring with spring
constant 1 pound/foot and is immersed
in a viscous medium with damping constant 2 pound-seconds/foot.  At time
$t=0$ the mass is pulled down
$1\over4$ foot from the equilibrium position and given an initial velocity
of one foot/second in the
downward direction.  Solve the spring equation and choose the best
description of what happens.  (g=32.)
\correct
The mass will move away from and then slowly return toward the equilibrium
position without ever passing
through it.
\wrong
The mass will pass through the equilibrium position once, and then slowly
return toward it.
\wrong
The mass will oscillate about the equilibrium position, moving the same
distance above and below it over
and over.
\wrong
The mass will pass through the equilibrium position exactly twice.
\wrong
The mass will oscillate about the equilibrium position, but with constantly
decreasing distance above and
below it.



\note %11
\problem
The expression \quad $-2\cos 5t+7\sin 5t$\quad can be rewritten in the form
\quad $R\cos(5t-\delta)$,\quad
where an approximate value for $\delta$ in radians is
\correct
1.849
\wrong
1.292
\wrong
4.990
\wrong
2.863
\wrong
-1.292


\note %12
\problem
For the equation \quad $(1-x^2)y''-6xy'-4y=0$,\quad find the first three
nonzero terms of the
series solution in powers of $x$ for which \quad $y(0)=1$\quad and \quad
$y'(0)=0$.
\correct
$1+2x^2+3x^4$
\wrong
$2+{5\over 3}x^3+{7\over3}x^5$
\wrong
$x^2+3x^4+4x^6$
\wrong
$1+x+2x^2$
\wrong
$1+x^2+{3\over 2}x^4$

\note %13
\problem
Find the recurrence relation for the series solution in powers of \quad
${\bf x-1}$\quad of the equation
$$(2x-x^2)y''-y=0.$$
\correct
$a_{n+2}={n^2-n+1\over n^2+3n+2}a_n$
\wrong
$a_{n+2}={n-1\over n+2}a_n$
\wrong
$a_{n+2}={(n-1)a_{n-1}+a_n\over (n+2)(n+1)}$
\wrong
$a_{n+2}={n+1\over n+2}a_n$
\wrong
$a_{n+2}={(n-1)a_{n-1}+na_n\over (n+2)(n+1)}$



\note %14
\problem
The equation \quad $(x-2)(x^2+4x+8)y''+(x^2+2)y'+(x-2)y=0$\hfil\break
has a power series solution in powers of $x+2$.  The general theory
guarantees that its radius of
convergence is at least
\correct
$2$.
\wrong
$2\sqrt 2$.
\wrong
$2\sqrt 5$.
\wrong
$4$.
\wrong

$3\sqrt 2$.

\note %15
\problem
The equation \quad $(x^2-1)^2y''+(x^2-1)y'+y=0$ \quad has
\correct
two regular singular points.
\wrong
two irregular singular points.
\wrong
one regular singular point and one irregular singular point.
\wrong
just one singular point, which is regular.
\wrong
just one singular point, which is irregular.

\note %16
\problem
Find the general solution of the Euler equation
$$x^2y''+7xy'+13y=0.$$
\correct
$x^{-3}(c_1\cos (2\ln|x|)+c_2 \sin(2\ln |x|))$
\wrong
$x^{3}(c_1\cos (\ln|x|)+c_2 \sin(\ln |x|))$
\wrong
$x^{-3}(c_1\cos (\ln2|x|)+c_2 \sin(\ln 2|x|))$
\wrong
$|x|^{-{7\over2}}(c_1\cos ({{\sqrt 3}\over 2}\ln|x|)+c_2 \sin({{\sqrt
3}\over 2}\ln |x|))$
\wrong
$|x|^{-{7\over2}}(c_1\cos (\ln{{\sqrt 3}\over 2}|x|)+c_2 \sin(\ln {{\sqrt
3}\over 2}|x|))$


 \end{document}

