%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam III}\quad{\sl April 21, 1992}\hfill Score:\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 15 multiple
choice questions worth 6 points each. You start with 10~points.

\def\problemspace{.25in}

%% PROBLEM
Let $L[y] = y'' + p(x)y' + q(x) y$. Determine which of the following
statements is FALSE.
%% CORRECT
If $y_1$ and $y_2$ are linearly independent solutions to $y'' =
f(x,y,y')$ then any solution to this differential equation can be written
as $c_1y_1 + c_2y_2$ for some choice of constants $c_1$ and $c_2$.
%% WRONG
If $y_1$ and $y_2$ are linearly independent solutions to $L[y] =
0$ then any solution to this differential equation can be written
as $c_1y_1 + c_2y_2$ for some choice of constants $c_1$ and $c_2$.
%% WRONG
$y_1$ and $y_2$ form a fundamental set of solutions to $L[y] = 0$ if and
only if $L[y_1] = 0$, $L[y_2] = 0$, and $W(y_1,y_2) \ne 0$.
%% WRONG
$y_1$ and $y_2$ are linearly independent functions of $x$ if and
only if $W(y_1,y_2) \ne 0$.
%% WRONG
If $y_1$ and $y_2$ are solutions to $L[y] = 0$ then $c_1y_1 + c_2y_2$ is
also a solution for any choice of constants $c_1$ and $c_2$.



%% PROBLEM
Compute the Wronskian of the functions $f(x) = \cos(x^2)$ and $g(x) =
\sin(x^2)$.
%% CORRECT
$2x$
%% WRONG
$4x$
%% WRONG
$2x(\cos(x^2) + \sin(x^2))$
%% WRONG
$2x(\sin(x^2) - \cos(x^2))$
%% WRONG
$1$


%% PROBLEM
Given that $y_1 = x^{-1}$ is a solution of
$$x^2 y'' + 3xy' + y = 0\qquad (x>0)$$
a second linearly independent solution can be found of the form $y_2 =
v y_1$ where $v$ is a non-constant function of $x$. Determine the
differential equation $v$ must satisfy.
%% CORRECT
$xv'' + v' = 0$
%% WRONG
$v' = x$
%% WRONG
$v'' - {1\over x}v' = 0$
%% WRONG
$v'' + x v' = 0$
%% WRONG
$v' = x^{-2}$


%% PROBLEM
Find the general solution of the homogeneous equation $y'' + 2y' + 4y = 0$.
%% CORRECT
$y = e^{-x}(c_1 \cos(\sqrt{3}x) + c_2 \sin(\sqrt{3}x))$
%% WRONG
$y = e^{x}( 3\cos(x) +  3\sin(x))$
%% WRONG
$y = e^{-x}(c_1 \cos(x) + c_2 \sin(x))$
%% WRONG
$y = e^{-2x}(c_1 \cos(3x) + c_2 \sin(3x))$
%% WRONG
$y = e^{\sqrt{3}x}(c_1 \cos(2x) + c_2 \sin(2x))$

%% PROBLEM
Suppose $y$ satisfies $4y'' - 4y' + y = 0$, $y(0) = 0$, and $y'(0) = 1$.
Find $y(1)$.
%% CORRECT
$\sqrt{e}$
%% WRONG
$e$
%% WRONG
$1/e$
%% WRONG
$1/\sqrt{e}$
%% WRONG
$e^2$


%% PROBLEM
Solve the initial value problem
$$3y'' + 2y' - y = 0,\quad y(0) = 1,\quad y'(0) = 0$$
%% CORRECT
$y = (3e^{x/3} + e^{-x})/4$
%% WRONG
$y = (\sqrt{3}e^{\sqrt{3}x} - 3 e^{x})/(\sqrt{3} - 3)$
%% WRONG
$y = (e^{x} + e^{-x})/2$
%% WRONG
$y = (\sqrt{2}e^{\sqrt{2}x} + 2e^{-x})/(\sqrt{2} - 2)$
%% WRONG
$y = (e^{-x} + 2e^{x/2})/3$


%% PROBLEM
Find a particular solution, $y_p$ to the differential equation
$$y'' - 2y' + 2y = x e^x$$
%% CORRECT
$y_p = x e^x$
%% WRONG
$y_p = (x+1)e^x$
%% WRONG
$y_p = (x^2 + x)e^x$
%% WRONG
$y_p = x^2 e^x$
%% WRONG
$y_p = xe^{x}\cos(x)$


%% PROBLEM
Determine the form of a particular solution to the differential equation
$$y'' - 2y' = x\cos(x) + e^{2x}$$
%% CORRECT
$y_p = (a_0x + a_1)\cos(x) + (b_0x + b_1)\sin(x) + Axe^{2x}$
%% WRONG
$y_p = (a_0x^2 + a_1x)\cos(x) + (b_0x^2 + b_1x)\sin(x) + Ae^{2x}$
%% WRONG
$y_p = (a_0x^2 + a_1x)\cos(x) + (b_0x^2 + b_1x)\sin(x) + (Ax+B)e^{2x}$
%% WRONG
$y_p = (a_0x^2 + a_1x)\cos(x) + Axe^{2x}$
%% WRONG
$y_p = (a_0x + a_1)e^{2x}(b_1\cos(x) + b_2\sin(x)) + Ae^{2x}$



%% PROBLEM
Let $y_p = u_1(x) x + u_2(x) e^x$ be a particular solution of
the differential equation
$$(1 - x)y'' + xy' - y = (x-1)^2 \qquad(x > 1)$$
given by the method of variation of parameters. Find $u_1$.
%% CORRECT
$u_1(x) = x$
%% WRONG
$u_1(x) = (x-1)^3/3$
%% WRONG
$u_1(x) = (x-1)e^x$
%% WRONG
$u_1(x) = x(x-1)$
%% WRONG
$u_1(x) = -(x-1)^2/2$



%% PROBLEM
A mass weighing 16~lbs stretches a spring 4~in. The mass is attached to a
viscous damper with damping constant $c$. Determine the value of $c$ if the
spring-mass system is to be critically damped (i.e., damped just enough to
eliminate oscillations).
%% CORRECT
$c = 4\sqrt{6}$
%% WRONG
$c = 4$
%% WRONG
$c = 4\sqrt{3}$
%% WRONG
$c = 8\sqrt{2}$
%% WRONG
$c = 8$



%% PROBLEM
Find the power series expansion for $f(x) = {1\over x^2}$ centered at $x_0 = 1$.
%% CORRECT 
$f(x) = 1 - 2(x-1) + 3(x-1)^2 - 4(x-1)^3 + \cdots$
%% WRONG 
$f(x) = 1 + 2(x-1) + 6(x-1)^2 + 24(x-1)^3 + \cdots$
%% WRONG 
$f(x) = 1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots$
%% WRONG 
$f(x) = 1 + {1\over2}x + {1\over3}x^2 + {1\over4}x + \cdots$
%% WRONG
$f(x) = 1 - x^2 + x^3 - x^4 + \cdots$


%% PROBLEM
Find the radius of convergence of the power series 
$\sum_{n=0}^\infty {3^n n^3\over n!} x^n$.
%% CORRECT
$\infty$
%% WRONG
$1/3$
%% WRONG
$1$
%% WRONG
$3$
%% WRONG
$\sqrt{3}$



%% PROBLEM
Determine a power series solution to the initial value problem
$$y'' - x^2 y = 0,\quad y(0) = 1,\quad y'(0) = 0$$
%% CORRECT
$y = 1 + {1\over 4\cdot3}x^4 + {1\over 8\cdot7\cdot4\cdot3} x^8 + \cdots$
%% WRONG
$y = 1 + {1\over 3\cdot2}x^3 + {1\over 6\cdot5\cdot3\cdot2} x^6 + \cdots$
%% WRONG
$y = 1 + {1\over 2!}x^2 + {1\over 4!}x^4 + \cdots$
%% WRONG
$y = 1 - {1\over2!}x^2 + {1\over 4!}x^4 - \cdots$
%% WRONG
$y = 1 - {1\over 4\cdot3}x^4 + {1\over 8\cdot 7}x^8 - \cdots$


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

%% PROBLEM
Determine a minimum for the radius of convergence of a power series
solution centered at $x_0 = 3$ to the differential equation
$$(x^2 + x) y'' - x y' + (x+1)y = 0$$
%% CORRECT
3
%% WRONG
1
%% WRONG
2
%% WRONG
$\infty$
%% WRONG
$0$

%% END
\bye
