\input ignoremce
\preface
\input epsf
\input calc
\def\problemskip{\vskip0.5in}
\def\ds{\displaystyle}

\title
\noindent
{\bf Math 226: Calculus IV}\hfill
Sign your Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\quad{\sl March 22, 1996}\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
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 or 
program capabilities.  Doing so is a violation of the Honor Code.
\bigskip\noindent
You are required to hand in the answer sheet and the problems.  You may tear off the 
blank sheets at the back for scratch work. {\bf Please tear off these scratch sheets
gently, one page at a time.}

\def\answerskip{\vskip0.25in}

\note %1
\problem
Determine the largest interval in which the given initial value problem is certain to have  a unique twice
differentiable solution.
 

\correct
$(0, 2)$
 

\wrong
$(- \infty,\infty)$
 

\wrong
$(0, 4)$
 

\wrong
$(2, \infty)$
 

\wrong
$(0, \infty)$




\note %2
\problem
Find the general solution of the differential equation
$$y^{\prime\prime} + (y^\prime)^2 = 0$$

\correct
$y = ln |x + c_1 | + c_2$

\wrong 
$y = {1\over 2} x^2 + c_1 x + c_2$ 


\wrong
$y = c_1 + c_2 e^{- x}$

\wrong
$y = c_1 cos 2 x + c_2 sin 2 x$

\wrong
$y = ln | e_1 x + c_2 |$



\note %3
\problem
Which of the following pair of functions are linearly  \underbar{dependent} on the interval $0 < x < \infty
?$

\correct
$e^x and e^{x-3}$

\wrong
$sin\  x\  and \ cos\ x$

\wrong
$ln\  x\  and \ ln\  (x + 2)$

\wrong
$e^x and\  e^{5x}$

\wrong
$x\  and\  x^2$


\note %4
\problem
Find the general solutions of the equation
\bigskip
$$y^{\prime\prime} - 2y^\prime  + 10 y = 20$$
\correct
$y = e^x (c_1 \ cos \ 3 x  +  c_2\  sin \ 3x) + 2$

\wrong
$y = e^{- x} (c_1\ cos\ 3x\ + c_2\ sin\ 3x ) - 4$

\wrong
$y = e^{3x} (c_1\ cos\ x + c_2\ sin\ x ) + 2$

\wrong
$y = c_1  e^x + c_2  e^{- 3 x} - 4$

\wrong
$y = e^x\ cos\ 3x + c^x sin\ 3x$


\note %5
\problem
Find $y (-1)$ if $y (x) $ is the solution to the initial value problem
\bigskip
$$y^{\prime\prime} - 4 y^\prime + 4 y = 0 , y (0) = 1, y^\prime  (0) = 3$$

\correct
$0$
\wrong
$2e^{- 2}$
\wrong
$3$
\wrong
$e^{-2}$
\wrong
$2 e^2$

\note %6
\problem
The function $ y_1 (x) = x^2 - 1$ is a solution to the equation
\bigskip
$$y^{\prime\prime} + y^\prime - {2\over x - 1}  \ y = 0$$
The function $y = y_1 |x| v |x|$  will be a record solution if $v|x|$ satisfies the differential equation
\correct
$(x^2 - 1) v^{P^{\prime\prime}} + (x^2 + 4 x - 1)\ v^1 - 0$
\wrong
$v^{\prime\prime} + v^\prime + v = 0$
\wrong
$v^{\prime\prime} + (x - 2)\ v^\prime = 0$
\wrong
$v^{\prime\prime} + (x^2 - 4 x)\ v^\prime - 0$
\wrong
There is no such $v (x)$.


\note %7
\problem
Find the general solution to the equation
$$y^{\prime\prime} - y^\prime  = - 2 x$$
\correct
$y = c_1 + c_2\  e^x + 2x + x^2$
\wrong
$y = c_1 e^x + c_2 x\  e^x + x^2 + 1$
\wrong
$y = c_1 x + c_2\ e^x + x^2$
\wrong
$y = c_1\ e^x + c_2 + 3x$
\wrong
$y = c_1 + e_2\ x e^x + 2x$


\note %8
\problem
The equation $y^{\prime\prime} + {1\over x} \ y^\prime - {1\over x^2} y = 0$ has solutions $y_1 = x$ and $y_2 = {1\over x}$ for $x > 0$.  Use the method of variation of parameters to find a solution of 
\bigskip
$$y^{\prime\prime} + {1\over x} y^\prime - {1\over x^2} y = {4\over x^3}$$
\correct
$y = 2 \ {2ln x\over x}$
\wrong
$y = - 4/x^2$
\wrong
$y = 2\ ln x$
\wrong
$y ^2 = x^3$
\wrong
$y = {1\over x}$


\note %9
\problem
A mechanical vibration is described by the differential equation
$$m\ \ u^{\prime\prime} + \gamma\  u^1 + k\  u = 0$$
where $m > 0,\ \gamma\ \geq\ 0,$ and $k > 0$.
Choose the one true statement.
\correct
If $\gamma ^2 - 4$ $ k m < 0,$ then all solutions $u |t|$ have the value $0$ for infinitely many $t$
\wrong
If $\gamma^2 - 4 k m = 0$, then are solutions $u (t)$ are periodic
\wrong
If $\gamma^2 - 4 k m >0$, then all solutions $u (t)$ have the value
\wrong
 0  for infinitley many  t 
\wrong
For a  certain choice of $\gamma$, all solutions $u (t)$ become unbounded as $t \longrightarrow \ \infty$. 
For a certain choice of $\sqrt {k\over m}$, all solutions $u (t)$ become unbounded as $t \longrightarrow
\infty$. 


\note %10
\problem
cos $wot +  \sqrt{3} sin\ wot$ \ is equal to   
\correct
$2 cos\ (wot - \pi/3)$
\wrong
$2 \cos (wo - {\pi\over3}) t$
\wrong
$\sqrt{3}\ cos (wot - \pi/6)$
\wrong
$2\ \cos (wot + \pi/6)$
\wrong
$\sqrt{3}\ cos (wot - \pi/3)$


\note %11
\problem
The coefficients of a power series solution $y |x| = \displaystyle\Sigma _{m=0}^\infty$ an ?  satisfy the
recurrence relation $an + 2 = - {1\over n = 2}$\  $a n$  for $\ge 0$.  Find the solution which fulfills
the initial condition $y (0) = 0 , y^\prime (0) = 1$ 
\correct
$x - 1\over3$  $\ x^3 + {1\over 5.3} x^5 - {1\over 7.5.3} x^7 + . . .$  
\wrong
$- {1\over3} x^3 + {1\over 5.3} x^5 - {1\over 7.5.3} x^7 + {1\over 9.7.5.3} x^9 + . . .$
\wrong
$x - {x^3\over 3} + {x^5\over 5} - {x^7\over 7} + . . .$ 
\wrong
$x - {1\over 3} x^2 + {1\over 5.3} x^3 - {1\over 7.5.3} x^4 + . . .$
\wrong
$1 - {x^2\over 3} + {x^4\over 5} - {x^6\over 7} + . . .$


\note %12
\problem
Using the method of undeterminied coefficients, find the correct from for a particular solution y of the
differential equation.
$$y{\prime\prime} + y\prime = e^2x + 5\  sin\  2x$$
\correct
$y = A e^{2x} + x (B \ cos\ 2x\ + C\ sin\ 2x)$
\wrong
$y = Axe^{2x} + Bx\ sin\ 2x$
\wrong
$y = Ae^{2x}\  + B\ cos\ 2x + C\ sin\ 2x$ 
\wrong
$y = Ae^{2x} + (Bx + C)  sin\ 2x$
\wrong
$Y = xe^{2x} (A\ cos\ 2x + B\ sin\ 2x)$


\note %13
\problem
The equation $(x^2 - 4x +5)\ y{\prime\prime} + x^3 y\prime - (4 x - 3) y = 0$ has a power series solution in
powers of $(x - 4)$.  From the general theory its radius of convergence is at least

\correct
$\sqrt 5$
\wrong
$- 2 + i$
\wrong
$\infty$
\wrong
$4$
\wrong
$5$


\note %14
\problem
The equation $y{\prime\prime} - 3x y\prime - 3y = 0$ has two linearly independent solutions expressed in
powers of x.  Find the recurrence relation that determines the coefficients
\correct
$an = 2 = {3 an\over n + 2}\ for \ n \ge$ 
\wrong
$an + 1 = {3 an\over n + 1}\ for\ n \ge 0$
\wrong
$an + 2 = \ -{an\over (n = 1)(n)}\ for\ n \ge 1\ and\ a_2 + a_0$.
\wrong
$an + 2 = {3 an + 1 + an\over n = 2}\ for\ n \ge 0$
\wrong
$an + 1 = {2 an\over n = 2}\ \ \ for\  \ n \ge 0$  


\note %15
\problem
Consider the singular points $x = 0\ and\ x = 1$ for the equation
$$x^3 (x - 1)^2 \ y{\prime\prime} +  (x^3 - x^2)\ y\prime \ + \ (x - 2)\  y = 0$$
\correct
$0$ is irregular and $1$ is regular. 
\wrong
Both are regular.
\wrong
Both are irregular.
\wrong
$0$ is regular and $1$ is irregular.
\wrong
Both are ordinary points.


\note %16
\problem
Find the general solution of the Euler equation
$$x^2 y{\prime\prime}\ - 5 x y\prime \ + \ 9 y = 0,\ x > 0$$
\correct
$x^3 (c_1 + c_2 \ ln\ x)$
\wrong
$x^{5/2}\ (c_1\ cos\ {3\over2}\ ln\ x)\ +\ c_2\ sin\ ({3\over2}\ ln\ x)$
\wrong
$c_1\ x^3\ +\ c_2\ x^4$
\wrong
$c_1\ cos ({3\over 2}\ ln\ x)\ +\ c_2\ sin\ ({3\over 2}\ ln\ x )$
\wrong
$c_1\ x^{5/2}\ +\ c_2\ x^{3/2}$



\end
