\def\problemspace{.25in}
\input calc
\let\ds=\displaystyle

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam III}\quad{\sl April 25, 1991}\hfill Score:\hbox to 2.5truein{\hrulefill}

 1in
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 15 multiple
choice questions worth 6 points each. You start with 10 points.


%% PROBLEM
Compute the Wronskian of the functions $e^x$ and $x e^x$.
%% CORRECT
{$e^{2x}$}
%% WRONG
{$e^{2x} + xe^{2x}$} 
%% WRONG
{$e^{x} + xe^{x}$} 
%% WRONG
{$e^{2x} - xe^{2x}$} 
%% WRONG
{$0$} 

%% PROBLEM
Determine which of the following pairs of functions are {\bf NOT}
linearly independent.
%% CORRECT
{$x\over\ds 1-x$, $1 - {1\over1-x}$  }
%% WRONG
{$\cos(x)$, $\sin(x)$}
%% WRONG
{$e^x$, $e^{2x}$} 
%% WRONG
{$x^2$, $x-1$}
%% WRONG
{$\cos(x)$, $\cos(3x)$}

%% PROBLEM
Given that $y_1 = x^3$ is a solution of
$$xy'' - y' - {3\over x}y = 0, \qquad x>0$$
find a second linearly independent solution of the form $y_2 = vy_1$ for some
non-constant function $v$.
%% CORRECT
{$y_2 = {1\over\ds x}$}
%% WRONG
{$y_2 = {1\over\ds x^4}$}
%% WRONG
{$y_2 = x^3e^{-{5\over4}x}$}
%% WRONG
{$y_2 = e^{-{5\over4}x}$}
%% WRONG
{$y_2 = x^3\ln(x)$}

%% PROBLEM
Find the general solution of
$$4y'' + 4y' + y = 0$$
%% CORRECT
{$y = e^{-{1\over2}x}(c_1 + c_2 x)$}
%% WRONG
{$y = e^{-{1\over2}x}(c_1 + c_2 \sqrt{x})$}
%% WRONG
{$y = c_1e^{-{1\over2}x} + c_2 \sqrt{x}$}
%% WRONG
{$y = c_1e^{-{1\over2}x} + c_2 x$}
%% WRONG
{$y = c_1e^{-x} + c_2 e^{-{1\over2}x}$}

%% PROBLEM
Solve the initial value problem
$$
y'' - 2y' - 3y = 0\qquad
y(0) = 0,\quad y'(0) = 1$$
%% CORRECT
{$y = {1\over4}e^{3x} - {1\over4}e^{-x}$}
%% WRONG
{$y = -{1\over4}e^{3x} + {1\over4}e^{x}$}
%% WRONG
{$y = {1\over3}e^{3x}$}
%% WRONG
{$y =  - e^{-x}$}
%% WRONG
{$y = {1\over3}e^{3x} - e^{x}$}



%% PROBLEM
Find the general solution to
$$2y'' + 10y' + 17y = 0$$
%% CORRECT
{$y = e^{-{5\over2}x}\left(c_1\cos({3\over2}x) + c_2\sin({3\over2}x)\right)$}
%% WRONG
{$y = c_1\cos({3\over2}x) + c_2\sin({3\over2}x)$}
%% WRONG
{$y = c_1e^{-{5\over2}x} + c_2e^{{3\over2}x}$}
%% WRONG
{$y = c_1e^{-2x}\cos(2x) + c_2e^{-2x}\sin(2x)$}
%% WRONG
{$y = c_1e^{-2x} + c_2xe^{-2x}$}


%% PROBLEM
Determine a suitable form for a particular solution $y_p$ of the equation
$$y'' + y' = x^2 + xe^x$$
%% CORRECT
{$y_p = x(Ax^2 + Bx + C) + (Dx + E)e^x$}
%% WRONG
{$y_p = (Ax^2 + Bx + C) + (Dx + E)e^x$}
%% WRONG
{$y_p = Ax^2 + x(Bx + C)e^x$}
%% WRONG
{$y_p = x^2(Ax^2 + Bx + C) + Dxe^x$}
%% WRONG
{$y_p = x(Ax^2 + Bx + C + Dxe^x)$}


%% PROBLEM
Find a particular solution $y_p$ to the differential equation
$$y'' - 3y' = \cos(x)$$
%% CORRECT
{$y_p = -{1\over10}\cos(x) - {3\over10}\sin(x)$}
%% WRONG
{$y_p = 1 + e^{3x}$}
%% WRONG
{$y_p = 1 + e^{3x} + \cos(x)$}
%% WRONG
{$y_p = -{1\over3}\cos(x) + {2\over3}\sin(x)$}
%% WRONG
{$y_p = {1\over15}\cos(x) + {8\over15}\sin(x)$}


%% PROBLEM
Let $y(x) = u_1(x)\cos(x) + u_2(x)\sin(x)$ be a particular solution of
$$y'' + y = \tan(x),\qquad 0 < x < {\pi\over2}$$
given by the method of variation of parameters. Find $u_2$.
%% CORRECT
{$u_2 = -\cos(x)$}
%% WRONG
{$u_2 = \sin(x)$}
%% WRONG
{$u_2 = \sec^2(x)$}
%% WRONG
{$u_2 = -\sec(x)$}
%% WRONG
{$u_2 = \cot(x)$}


%% PROBLEM
A 3~lb weight stretches a spring 6~inches. The weight is hit from below giving
it an upward velocity of~1 ft/sec.  Find the resulting
displacement~$u$ of the weight from equilibrium.
Assume the positive $u$ direction is {\sl down} and neglect air resistance.
%% CORRECT
{$u = -{1\over 8}\sin(8t)$}
%% WRONG
{$u = {1\over 8}\cos(8t)$}
%% WRONG
{$u = -{1\over 4\sqrt{3}}\sin(4\sqrt{3}t)$}
%% WRONG
{$u = {1\over 4\sqrt{3}}\cos(4\sqrt{3}t)$}
%% WRONG
{$u = -\cos(8t) + \sin(8t)$}


%% PROBLEM
Which of the following is the power series for $e^x$ centered at~$x_0 = 1$:
%% CORRECT
{$e + e(x-1) + {e\over2!}(x-1)^2 + {e\over3!}(x-1)^3 + \ldots$}
%% WRONG
{$1 + (x-1) + {1\over2!}(x-1)^2 + {1\over3!}(x-1)^3 + \ldots$}
%% WRONG
{$1 + x + {1\over2!}x^2 + {1\over3!}x^3 + \ldots$}
%% WRONG
{$e - e(x-1) + {e\over2!}(x-1)^2 - {e\over3!}(x-1)^3 + \ldots$}
%% WRONG
{$1 - x + {1\over2!}x^2 - {1\over3!}x^3 + \ldots$}


%% PROBLEM
Determine the radius of convergence of the series
$$\sum_{n=0}^\infty (-1)^n{1\over 4^{n+1}}x^{2n}$$
%% CORRECT
{$2$}
%% WRONG
{$4$}
%% WRONG
{$1$}
%% WRONG
{${1\over4}$}
%% WRONG
{$\infty$}


%% PROBLEM
Find the recurrence relation for the coefficients of a power series
solution centered at~$x_0 = 0$ of the differential equation
$$(1-x)y'' + y = 0$$
%% CORRECT
{$a_{n+2} = {n\over n+2}a_{n+1} - {1\over (n+2)(n+1)}a_n$}
%% WRONG
{$a_{n+2} = {n\over n+2}a_{n+1} - {1\over n+2}a_n$}
%% WRONG
{$a_{n+2} = {n\over (n+2)(n+1)}a_n$}
%% WRONG
{$a_{n+2} = {1\over (n+2)(n+1)}a_{n+1} - {1\over (n+2)(n+1)}a_n$}
%% WRONG
{$a_{n+2} = {n\over (n+2)(n+1)}a_{n+1} - {1\over n+1}a_n$}


%% PROBLEM
Determine a lower bound for the radius of convergence of a power series solution
centered at $x = 2$ of the differential equation
$$(x + 3)^2y'' + xy' - y = 0$$
%% CORRECT
{5}
%% WRONG
{4}
%% WRONG
{3}
%% WRONG
{2}
%% WRONG
{1}


%% PROBLEM
Find a power series solution to the initial value problem
$$
y'' + xy' = 0\qquad
y(0) = 0,\quad y'(0) = 1
$$
%% CORRECT
{$y = x - {1\over6}x^3 + {1\over40}x^5 - {1\over336}x^7 + \ldots$}
%% WRONG
{$y = x - {1\over3}x^3 + {1\over5}x^5 - {1\over7}x^7 + \ldots$}
%% WRONG
{$y = x + {1\over6}x^2 + {1\over20}x^3 +- {1\over35}x^4 + \ldots$}
%% WRONG
{$y = x - {1\over4}x^2 + {1\over75}x^3 - {1\over116}x^5 + \ldots$}
%% WRONG
{$y = x - {1\over12}x^3 + {1\over105}x^5 - {1\over286}x^7 + \ldots$}


%% END
\end
