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{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam III}\quad{\sl April 24, 1990}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip
\leavevmode\hfill Section:\hbox to 2.5truein{\hrulefill}

\bigskip
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 14
questions worth 7 points.
\bigskip\bigskip

{
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  \answersheet{1}{14}{1}{2}{\theanswers{}{}{}{}{}}
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}

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\pageno=1
\footline={\hfil{\rm \folio}\hfil}
\bigskip\bigskip

\newproblem
Given that $y_1 = x^2$ is a solution of
$$2x^2y'' - 3xy' + 2y = 0$$
a second independent solution can be found of the form $y_2 = v y_1$ where $v$ is
a non-constant function. Determine the differential equation $v$ must satisfy.
\answers
{$v''x^2 + 4xv' + 2v = 0$}
{$v'x^2 + 2xv = 0$\endanswerline}
{$v''x^2 + v'(4x + x^2) = 0$}
{$2xv'' + 5v' = 0$\endanswerline}
{{\sl None of the above}}

\newproblem
Consider the differential equation
$$4xy'' - 8y' + {9\over x}y = 0, \qquad x > 0$$
with solution $y_1 = x^{3/2}$. A second solution is given by $y_2 = v y_1$
where $v$ satisfies
$$xv'' + v' = 0$$
Find $y_2$.
\answers
{$y_2 = -x^{1/2}$}
{$y_2 = x^{3/2}-\ln(x)$}
{$y_2 = -x^{-1}$\endanswerline}
{$y_2 = -x^{-1/2}$}
{$y_2 = x^{3/2}\ln(x)$}

\newproblem
Find the general solution of the homogeneous equation
$$y'' - 3y' - 4y = 0$$
\answers
{$y = c_1 e^{4x} + c_2 e^{-x}$}
{$y = c_1 e^{-3x} + c_2 e^{-4x}$}
{$y = c_1 e^{-4x} + c_2 e^{x}$\endanswerline}
{$y = c_1 e^{4x}\cos(x) + c_2 e^{4x}\sin(x)$}
{$y = c_1 \cos{4x} + c_2 \sin{x}$}

\newproblem
Find the general solution of the homogeneous equation
$$y'' + 2 y' + 2y = 0$$
\answers
{$y = c_1 e^{-x}\cos(x) + c_2 e^{-x}\sin(x)$}
{$y = c_1 e^{x}\cos(x) + c_2 e^{x}\sin(x)$}
{$y = c_1 e^{2x} + c_2 e^{-x}$\endanswerline}
{$y = c_1 e^{2x} + c_2 x e^{2x}$}
{$y = c_1 e^{-x} + c_2 x e^{-x}$}

\newproblem
Solve the initial value problem
$$y'' - 2 y' + y = 0,\qquad y(0) = 1,\quad y'(0) = 3$$
\answers
{$y = e^{3x}$}
{$y = 2e^x - e^{-x}$}
{$y = e^x + 2x e^x$\endanswerline}
{$y = e^x + 2x$}
{{\sl None of the above}}

\newproblem
Use the method of undetermined coefficients to find a particular solution and then
solve the initial value problem
$$y'' - y = e^{2x}, \qquad y(0) = y'(0) = 0$$
\answers
{$y = {1\over2}e^x - {3\over2}x e^{-x} + {1\over2}e^{2x}$}
{$y = -{3\over4}e^x + {1\over4}e^{-x} + {1\over2} e^{2x}$}
{$y = 0$\endanswerline}
{$y = -{1\over3}e^x - {x\over 3} e^x  - {1\over3}e^{2x}$}
{$y = {1\over6}e^{-x} - {1\over2}e^{x} + {1\over3}e^{2x}$}

\newproblem
Using the method of undetermined coefficients, find the correct form for a
particular solution $y_p$ in the differential equation
$$y'' + 10y' + 25y = x^2 + e^x$$
\answers
{$y_p = x(Ax^2 + Bx + C) + De^x$}
{$y_p = ax^2 + Bx + C + De^{-5x} + Exe^{-5x}$\endanswerline}
{$y_p = Ax^2 + Be^x$}
{$y_p = Ax^2 + Bx + C + De^x$\endanswerline}
{$y_p = Ax^2 + Bx + C + Dxe^x$}


\newproblem
Let $y(x) = u_1(x)\cos(x) + u_2(x)\sin(x)$ be the solution of
$$y'' + y = \sin(x)\cos(x)$$
given by the method of variation of parameters. Find $u_1$.
\answers
{$u_1 = \tan(x)$}
{$u_1 = -\cot(x)$}
{$u_1 = -\sin^2(x)\tan(x)$\endanswerline}
{$u_1 = -{1\over3}\sin^3(x)$}
{$u_1 = {1\over2}\cos^2(x)$}

\newproblem
The displacement in a certain spring-mass system with no external force and no
damping satisfies the initial value problem
$$m\ddot u + k u = 0, \qquad u(0) = u_0 (\ne 0),\quad\dot u(0) = 0$$
For what values of time $T$ is the displacement zero?
\answers
{$T = (2j+1)\sqrt{m\over k}{\pi\over2},\quad j = 0,1,2,\ldots$}
{$T = (2j+1){\pi\over2 u_0},\quad j = 0,1,2,\ldots$\endanswerline}
{$T = (2j+1)\sqrt{m\over k u_0}\pi,\quad j = 0,1,2,\ldots$}
{$T = 2j\sqrt{m\over k}{1\over u_0},\quad j = 0,1,2,\ldots$\endanswerline}
{$T = j\sqrt{m\over k}{\pi},\quad j = 0,1,2,\ldots$}

\newproblem
Find the radius of convergence of the power series
$$\sum_{n=0}^\infty {n^2\over 2^n} x^n$$
\answers
{0} 
{1/2}
{1}
{2}
{$\infty$}

\newproblem
Compute the Taylor series for $f(x) = {1\over2-x}$ about $x_0 = 1$.
\answers
{$1 + (x-1) + (x-1)^2 + \ldots$}
{${1\over2} + {1\over2^2} (x-1) + {1\over2^3} (x-1)^2 + \ldots$\endanswerline}
{$1 - x + x^2 - x^3 + \ldots$} 
{$1 - (x-1) + (x-1)^2 - (x-1)^3 + \ldots$\endanswerline}
{${1\over2} + {1\over2^2} x + {1\over2^3} x^2 + \ldots$}

\newproblem
Assuming $a_0 =1$, determine $a_n$ for $n > 1$ so that
$$\sum_{n=1}^\infty{n a_n x^{n-1}} - 3\sum_{n=0}^\infty {a_n x^n} = 0$$

\answers
{$a_n = {1\over n!}$}
{$a_n = {3\over n}$}
{$a_n = {3\over n!}$}
{$a_n = {3^n\over n}$} 
{$a_n = {3^n\over n!}$}


\newproblem
Find the recurrence relation for the coefficients of a power series solution about
$x_0 = 0$ of the differential equation $y'' + x y' + y = 0$.
\answers
{$a_{n+2} = {1\over n}a_{n}$}
{$a_{n+2} = -{1\over n+2}a_{n}$} 
{$a_{n} = -{1\over n+2}a_{n+2}$\endanswerline}
{$a_{n+2} = -{1\over (n+2)(n+1)}a_{n}$}
{$a_n = {1\over (n+2)(n+1)}a_{n+2}$}


\newproblem
Determine the lower bound for the radius of convergence of a power series solution
about $x_0 = -1$ of the differential equation $(x^2 + 2x - 3) y'' - y = 0$.
\answers
{$1/\sqrt{3}$}
{1} 
{2}
{$\sqrt{3}$}
{$\infty$}




\bye