\input calc
\def\problemspace{0.1in}

%% TITLE
\noindent
{\bf Math 226: Calculus IV}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl May 5, 1993}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\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 25 multiple choice questions worth 6 points each.

\def\answerspace{0.3in}


%% PROBLEM
Let $\X_1 = (1,1,1,1)$, $\X_2 = (2,0,3,5)$, $\X_3 = (0,1,0,2)$,
and $\B = (1,-3,2,0)$. Find constants $c_1$, $c_2$, and $c_3$ such
that $c_1\X_1 + c_2\X_2 + c_3\X_3 = \B$. The value of $c_3$ is:

%% CORRECT
$-2$
%% WRONG
$1$
%% WRONG
$-1$
%% WRONG
$3$
%% WRONG
${1\over2}$


%% PROBLEM
Apply the Gram-Schmidt process to the vectors $\X_1 = (1,0,1,0)$,
$\X_2 = (0,1,0,1)$ and $\X_3 = (0,0,1,1)$ to obtain vectors $\Q_1$,
$\Q_2$, $\Q_3$.  The vector $\Q_3$ is:

%% CORRECT
$(-{1\over2},-{1\over2},{1\over2},{1\over2})$
%% WRONG
$({4\over5\sqrt{2}}, -{3\over5\sqrt{2}}, -{4\over5\sqrt{2}}, {3\over5\sqrt{2}})$
%% WRONG
$({1\over\sqrt{2}},0,-{1\over\sqrt{2}},0)$
%% WRONG
$(0,-{1\over\sqrt{2}},0,{1\over\sqrt{2}})$
%% WRONG
$(0,0,{1\over\sqrt{2}},{1\over\sqrt{2}})$


%% PROBLEM
Calculate the determinant of the matrix
$$\bmatrix{1&0&2&0&3\cr0&4&0&5&0\cr6&0&7&0&8\cr0&9&0&10&0\cr11&0&12&0&13\cr}$$

%% CORRECT
$0$
%% WRONG
$546$
%% WRONG
$840$
%% WRONG
$336$
%% WRONG
$945$

%% PROBLEM
Let $A$ be the matrix
$$A = \bmatrix{1&1&1&1\cr0&1&1&1\cr0&0&1&1\cr0&0&0&1\cr}$$
Calculate the
entry in the first row and fourth column of the inverse matrix,
$A_{(1,4)}^{-1}$.

%% CORRECT
$0$
%% WRONG
$1$
%% WRONG
$-1$
%% WRONG
$1/2$
%% WRONG
$-1/4$


%% PROBLEM
The differential equation
$$(3y^2 - 4x(y^3 + 1))\,dx + xy(2 - 3xy)\,dy = 0$$

%% WRONG
is exact.
%% WRONG
is homogenous.
%% CORRECT
has an integrating factor that is a function of $x$ alone.
%% WRONG
has an integrating factor that is a function of $y$ alone.
%% WRONG
{\sl none of the above}
%% STET

%% PROBLEM
A water droplet falls from rest in a cloud and accumulates water
at a constant rate on its way down so that its mass is given by $m
= m_0 + \alpha t$ where $m_0$ is its initial mass and $\alpha$ is
a constant. Assume that the force of gravity is $-m g$ and that
the frictional force due to air resistance is given by $-c v$ where
$v$ is the velocity of the droplet and $c$ is a constant. Derive
an expression for $v$ as a function of $t$. (The upward direction
is positive.)

%% CORRECT
$v = -{g\over c + \alpha}\big(m_0 + \alpha t - m_0^{1+c/\alpha}(m_0 + \alpha t)^{- c/\alpha}\big)$
%% WRONG
$v = -{ m_0 g\over c} (1 - e^{-ct/m_0})$
%% WRONG
$v = -{\alpha g\over c}\Big[t + {m_0\over\alpha} - {1\over c} - e^{-ct}\big({m_0\over\alpha} - {1\over c}\big)\Big]$
%% WRONG
$v = -{m_0 g\over c}\big(1 - {\alpha\over c}e^{-ct/\alpha}\big)$
%% WRONG
$v = m_0^{c/\alpha-1}-{g\over m_0}(m_0 + \alpha t)^{c/\alpha}$


%% PROBLEM
Find the general solution to the differential equation
$$y'' + 2\sqrt{2}y' + 2y = 0$$

%% CORRECT
$y = c_1e^{-\sqrt{2}x} + c_2 x e^{-\sqrt{2}x}$
%% WRONG
$y = c_1e^{x/\sqrt{2}} + c_2 e^{-x/\sqrt{2}}$
%% WRONG
$y = c_1e^{\sqrt{2}x} + c_2 e^{-\sqrt{2}x}$
%% WRONG
$y = c_1e^{-x/\sqrt{2}} + c_2 x e^{-x/\sqrt{2}}$
%% WRONG
$y = c_1e^{\sqrt{2}x} + c_2 x e^{\sqrt{2}x}$


%% PROBLEM
Let $y(x)$ be the solution to the initial value problem
$$y' + {2\over x} y = x,\quad y(1) = 0$$
Find $y(2)$.

%% CORRECT
$y = {15\over 16}$
%% WRONG
$2\ln(2)$
%% WRONG
${3\over4}$
%% WRONG
${7\over8}$
%% WRONG
${1\over2}$


%% PROBLEM
Classify the equilibrium solutions to the differential equation
$${dN\over dt} = N (N^2 - 4N + 3)$$

%% CORRECT
$N = 0$ unstable, $N = 1$ stable, $N = 3$ unstable
%% WRONG
$N = 0$ unstable, $N = 1$ unstable, $N = 3$ stable
%% WRONG
$N = 0$ unstable, $N = 1$ stable, $N = 3$ stable
%% WRONG
$N = 0$ stable, $N = 1$ unstable, $N = 3$ unstable
%% WRONG
$N = 0$ stable, $N = 1$ stable, $N = 3$ unstable


%% PROBLEM
Determine the largest region in the plane where the existence of
a unique solution through any point is guaranteed for the differential
equation $y' = (x y)^{1/3}$

%% CORRECT
$\{(x,y)\,|\, y \ne 0\}$
%% WRONG
$\{(x,y)\,|\, x\ne 0\ {\rm and}\ y \ne 0\}$
%% WRONG
$\{(x,y)\,|\, x>0\ {\rm and}\ y>0,\ {\rm or}\ x<0\ {\rm and}\ y<0\}$
%% WRONG
$\{(x,y)\,|\, y>0\}$
%% WRONG
{\sl the entire plane}


%% PROBLEM
The solution of the differential equation
$$(3x^2y+ y^2)\, dx + (x^3 + 2xy - 1)\, dy = 0$$
is given implicitly by:

%% CORRECT
$x^3 y +  x y^2 - y = c$
%% WRONG
$x^3y +  x y^2 - y$
%% WRONG
$x^3 + {1\over 3}y^3 + x y^2 = c$
%% WRONG
$x^3 + {1\over 3}y^3 + xy^2$
%% WRONG
$x^3 y + x y^2 = c$


%% PROBLEM
Solve the differential equation
$$y' = {y^5 + x^5\over x y^4}\quad (x > 0)$$
by making the substitution $v = {y\over x}$.

%% CORRECT
$y = x(5\ln(x) + c)^{1/5}$
%% WRONG
$y = (x^5 + c\ln(x))^{1/5}$
%% WRONG
$y = (x^4 + c)^{1/4}$
%% WRONG
$y = c e^{x^5}$
%% WRONG
$y = \Big(c\ln(x) + {1\over x^5}\Big)^{1/5}$


%% PROBLEM
The water bath surrounding a certain nuclear reactor contains
radioactive material that decays with a half-life of 3~days. New
radioactive material is added to the bath at the constant rate of
1~gram per day. Assume the bath contains no radioactive material
initially.  Calculate an expression for the amount of radioactive
material, $u(t)$ (grams), in the bath as a function of~$t$ (days).

%% CORRECT
$u(t) = {3\over \ln{2}}\big(1 - \big({1\over2}\big)^{t/3}\big)$
%% WRONG
$u(t) = 1 - \big({1\over2}\big)^{-t/3}$
%% WRONG
$u(t) = {\ln{3}\over2}\big(\big({1\over2}\big)^{t/3} - 1\big)$
%% WRONG
$u(t) = \big({1\over2}\big)^{t/3} - 1$
%% WRONG
$u(t) = t - \big({1\over2}\big)^{-t/3}$


%% PROBLEM
Determine which of the following is {\bf not} a linear differential equation.

%% CORRECT
$2y'' + 3y' + 4y = x/y$
%% WRONG
$y = x y''$
%% WRONG
$\sin(x) y''  = {\cos(x) + y\over x^2}$
%% WRONG
$e^x y' + \ln(x) y - 17 x^2 = 0$
%% WRONG
$xy'' + {1\over x}y = x^3$


%% PROBLEM
Compute the Wronskian of the functions
$$f(x) = {1\over x - 2}\quad {\rm and}\quad g(x) = 1 - {x\over x - 2}$$

%% CORRECT
$0$
%% WRONG
${1\over(x - 2)^2}$
%% WRONG
$-{2\over(x - 2)^3}$
%% WRONG
$-{2x\over(x - 2)^3}$
%% WRONG
${x\over(x - 2)^2}$

%% PROBLEM
Solve the differential equation with initial conditions
$$y'' + 2y' + 4y = 0,\quad y(0) = 1,\quad  y'(0) = 0$$

%% CORRECT
$y = e^{-x}(\cos(\sqrt{3}x) +{\sqrt{3}\over3}\sin(\sqrt{3}x))$
%% WRONG
$y = e^{-x}(\cos(2x) +{1\over2}\sin(2x))$
%% WRONG
$y = e^{x}(\cos(3x) -{1\over3}\sin(3x))$
%% WRONG
$y = e^{x/2}(\cos(\sqrt{2}x) +{\sqrt{2}\over4}\sin(\sqrt{2}x))$
%% WRONG
$y = e^{-x/2}(\cos(x) +{1\over2}\sin(x))$

%% PROBLEM
Given that $y_1 = x$ is a solution of the differential equation
$$x^2 y'' - x(x+2) y' + (x+2) y = 0,\quad x>0$$
a second linearly independent solution has the form $y_2 = v y_1$
for some non-constant function $v$. Determine the dif\-fer\-en\-tial
equa\-tion that $v$ must satisfy.

%% CORRECT
$v'' = v'$
%% WRONG
$x^2v'' = 2(x+1)v'$
%% WRONG
$x^3v'' = (x^2 + 2)v'$
%% WRONG
$v'' = (x^2+1)v'$
%% WRONG
$v'' = (x+1)v'$

%% PROBLEM
Determine the form of a particular solution, $y_p$, to the differential
equation $$y'' + y = x^2 + \cos(x)$$ given by the method of
undetermined coefficients.

%% CORRECT
$y_p = (Ax^2 + Bx + C) + x(D\cos(x) + E\sin(x))$
%% WRONG
$y_p = x(Ax^2 + Bx + C) + (D\cos(x) + E\sin(x))$
%% WRONG
$y_p = Ax^2 + B\cos(x) + C\sin(x)$
%% WRONG
$y_p = Ax^2 + Bx\cos(x) + Cx\sin(x)$
%% WRONG
$y_p = x(Ax^2 + Bx + C) + x(D\cos(x) + E\sin(x))$

%% PROBLEM
A particular solution to the differential equation
$$y'' - 2y' + y = {e^x\over 1+x^2}$$
given by the method of variation of parameters has the form $y_p
= u_1 y_1 + u_2 y_2$ where $y_1 = e^x$ and $y_2 = xe^x$ are solutions
of the corresponding homogeneous equation.  Determine $u_2$.

%% CORRECT
$\tan^{-1}(x)$
%% WRONG
$\ln \sqrt{1+x^2}$
%% WRONG
$\sin^{-1}(x)$
%% WRONG
$e^x\ln\sqrt{1+x^2}$
%% WRONG
$e^x\tan^{-1}(x)$

%% PROBLEM
Determine the  radius of convergence of a power series solution
centered at $x_0 = 0$ to the differential equation
$$(x^2 + 4)y'' + y' - x^2 y = 0$$

%% CORRECT
$2$
%% WRONG
$1$
%% WRONG
$3$
%% WRONG
$4$
%% WRONG
$\infty$

%% PROBLEM
Compute the Taylor series expansion for $f(x) = {1\over (1-x)^3}$
centered at $x_0 = 0$.

%% CORRECT
$y = 1 + 3x + 6x^2 + 10x^3 + 15x^4 + \cdots$
%% WRONG
$y = 1 + 3x + 12x^2 + 60x^3 + 720x^4 + \cdots$
%% WRONG
$y = 1 + 3x + 4x^2 + 5x^3 + 6x^4 + \cdots$
%% WRONG
$y = 1 + 3x + 15x^3 + 105x^5 + 945x^4 + \cdots$
%% WRONG
$y = 1 + 3x^3 + 6x^6 + 9x^9 + 12x^{12} + \cdots$

%% PROBLEM
Find a power series solution centered at $x_0 = 0$ to the differential
equation $$y'' - y' - 2xy = 0$$ that satisfies the initial conditions
$y(0) = 1$, and $y'(0) = 0$.

%% CORRECT
$y = 1 + {1\over3}x^3 + {1\over12}x^4 + {1\over60}x^5 + \cdots$
%% WRONG
$y = 1 + x + {1\over2}x^2 + {2\over3}x^3 + {1\over3}x^4 + \cdots$
%% WRONG
$y = 1 + {1\over2}x^2 + {1\over12}x^3 + {1\over24}x^4 + \cdots$
%% WRONG
$y = 1 + {1\over4}x^3 + {1\over20}x^4 + {1\over45}x^5 + \cdots$
%% WRONG
$y = 1 + {1\over6}x^2 + {1\over24}x^3 + {1\over60}x^4 + \cdots$

%% PROBLEM
Find a fundamental set of solutions for the Euler equation
$x^2y'' + 3xy' + 10y = 0$

%% CORRECT
$\left\{{\cos(3\ln|x|)\over x}, {\sin(3\ln|x|)\over x}\right\}$
%% WRONG
$\left\{{1\over x}, {1\over x^3}\right\}$
%% WRONG
$\left\{{\cos(\sqrt{31}\ln|x|)\over x^{3/2}},
{\sin(\sqrt{31}\ln|x|)\over x^{3/2}}\right\}$
%% WRONG
$\left\{{1\over x^2}, {1\over x^5}\right\}$
%% WRONG
$\left\{{1\over\sqrt{x}}, {\ln|x|\over\sqrt{x}}\right\}$

%% PROBLEM
The following differential equation has a regular singular point
at $x_0 = 0$:  $$x^2 y'' + 6x y' + (6 + x^2) y = 0$$
Determine which of the power series below is a solution to this equation.

%% CORRECT
$y = x^{-2}(1 - {1\over6} x^2 + {1\over 120} x^4 - \cdots)$
%% WRONG
$y = x^{-2}(1 + {1\over6} x^3 + {1\over 120} x^5 + \cdots)$
%% WRONG
$y = x^{-3}(1 - {1\over2} x^2 + {1\over 12} x^4 - \cdots)$
%% WRONG
$y = x^{-3}(1 + {1\over2} x^3 + {1\over 12} x^5 + \cdots)$
%% WRONG
$y = x^{-1}(1 - {1\over12} x^2 + {1\over 360} x^4 - \cdots)$

%% PROBLEM
The differential equation
$$ x^2 y'' + (e^x - 1) y' - 2 y = 0$$
is guaranteed to have a solution of the form:

%% CORRECT
$y = |x|^{\sqrt{2}}\left(1 + \sum_{n=0}^\infty a_n x^n\right)$
%% WRONG
$y = |x|^2\left(1 + \sum_{n=0}^\infty a_n x^n\right)$
%% WRONG
$y = |x|^{-2}\left(1 + \sum_{n=0}^\infty a_n x^n\right)$
%% WRONG
$y = \left(1 + \sum_{n=0}^\infty a_n x^n\right)$
%% WRONG
$y = |x|^r$ for some $r$

%% END
\end
