\input calc
\def\problemspace{.25in}

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam II}\quad{\sl March 21, 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
Find the general solution of the differential equation
$$x y' + 2y = \sin(x)$$
%% CORRECT
{$y = {1\over x^2}[\sin(x) - x\cos(x) + c]$}
%% WRONG
{$y = {1\over x}[\sin(x) + x\cos(x) + c]$} 
%% WRONG
{$y = {1\over x^2}[\sin(x) + \cos(x) + c]$} 
%% WRONG
{$y = cx[\sin(x) - x\cos(x)]$} 
%% WRONG
{$y = cx^2[\sin(x) - x\cos(x)]$} 

%% PROBLEM
Suppose $y(0) = 0$ and
$$y' - y = x^3 e^x$$
Find $y(1)$.
%% CORRECT
{$e/4$}
%% WRONG
{$e$}
%% WRONG
{$1$} 
%% WRONG
{$4e$}
%% WRONG
{$e^{-1}$}

%% PROBLEM
Find the region in the $xy$-plane where there exists a unique solution
through any specified point for the differential equation
$$y' = (x^2 + y^2 - 1)^{1/3}$$
%% CORRECT
{$\{(x,y)\,|\, x^2 + y^2 < 1 {\rm\ and\ }x^2 + y^2 > 1\}$}
%% WRONG
{$\{(x,y)\,|\, x^2 + y^2 < 1\}$}
%% WRONG
{$\{(x,y)\,|\, x^2 + y^2 > 1\}$} 
%% WRONG
{$\{(x,y)\,|\,y^2 \ne 0\}$}
%% WRONG
{{\sl everywhere}}

%% PROBLEM
On what interval is the solution to the following initial value problem valid?
$$y' = -2{x\over y}\qquad y(0) = 1$$
%% CORRECT
{$-{\sqrt{2}\over 2} < x < {\sqrt{2}\over 2}$}
%% WRONG
{$x < {1\over 2}$}
%% WRONG
{$-{1\over 2} < x < {1\over 2}$}
%% WRONG
{$x > 0$}
%% WRONG
{$x  < {\sqrt{2}\over 2}$}

%% PROBLEM
Determine the integral curve of the differential equation
$${dy\over dx} = y + {1\over y}$$
which passes through the point $(0,2)$.
%% CORRECT
{$5e^{2x} - y^2 = 1$}
%% WRONG
{$y^2 = 2x + 4$}
%% WRONG
{$\ln|y^2+1| = {1\over 2}x^2 + \ln(5)$}
%% WRONG
{${1\over2}y^2 + \ln(y) = x + 2 + \ln(2)$}
%% WRONG
{$\sqrt{y^2 + 1} = x + \sqrt{5}$}



%% PROBLEM
A bank offers two types of certificates of deposit. The first type pays
interest at 7\% per year compounded continuously. The second type pays interest
at 8\% per year, but requires an initial fee of 2\% deducted from the amount
deposited. Suppose an equal amount of money is deposited in each type of
certificate. At what time will the two certificates be worth the same amount?
%% CORRECT
{$2.02$ years}
%% WRONG
{$0$ years}
%% WRONG
{$1.57$ years}
%% WRONG
{$3.11$ years}
%% WRONG
{$0.98$ years}


%% PROBLEM
The water in a 200 gal aquarium is to be replenished by first draining out 100
gal and then refilling the aquarium with fresh water at a rate of 2 gal/min. At
the same time, the mixture continues to drain out of the tank at a rate of 1
gal/min. If the aquarium originally had 5\% impurities in the water, what will
the concentration of impurities be when the the tank is full?
%% CORRECT
{$1.25\%$}
%% WRONG
{$0.05\%$}
%% WRONG
{$2.71\%$}
%% WRONG
{$0.47\%$}
%% WRONG
{$3.52\%$}


%% PROBLEM
Classify the equilibrium solutions of the logistic equation
$${dN\over dt} = 100 - 25N + N^2$$
%% CORRECT
{\vtop{\hbox{\strut$N=5$ stable}
\hbox{\strut$N=20$ unstable}}}
%% WRONG
{\vtop{\hbox{\strut$N=0$ stable}
\hbox{\strut$N=10$ unstable}}}
%% WRONG
{\vtop{\hbox{\strut$N=5$ unstable}
\hbox{\strut$N=20$ stable}}}
%% WRONG
{\vtop{\hbox{\strut$N=0$ unstable}
\hbox{\strut$N=5$ stable}
\hbox{\strut$N=20$ unstable}}}
%% WRONG
{\vtop{\hbox{\strut$N=0$ unstable}
\hbox{\strut$N=10$ unstable}
\hbox{\strut$N=20$ stable}}}


%% PROBLEM
A 12 kg body falls from rest in a medium offering a resistance equal to 0.75
times the square of the velocity. Find the limiting velocity of the body.
%% CORRECT
{12.52 m/sec}
%% WRONG
{10.02 m/sec}
%% WRONG
{9.13 m/sec}
%% WRONG
{8.66 m/sec}
%% WRONG
{7.41 m/sec}


%% PROBLEM
Solve the differential equation
$$(3x^2 - 2xy + 2)\,dx + (3y^2 - x^2 + 1)\,dy = 0$$
subject to the initial condition $y(0) = 1$.
%% CORRECT
{$x^3 + y^3 - x^2 y + 2x + y = 2$}
%% WRONG
{$f(x,y) = x^3 + y^3 - x^2 y + 2x + y$}
%% WRONG
{$x^3 - x^2 y + 2x = 0$}
%% WRONG
{$f(x,y) = x^3 - x^2 y + 2x$}
%% WRONG
{$y^3 - x^2y + y = 2$}


%% PROBLEM
Find an integrating factor for the differential equation
$$(y + e^{y-x})\,dx + (1 + xe^{y-x})\,dy$$
%% CORRECT
{$e^x$}
%% WRONG
{$x$}
%% WRONG
{$e^{-y}$}
%% WRONG
{$xe^x$}
%% WRONG
{$1 - e^{-y}$}


%% PROBLEM
Solve the differential equation
$${dy\over dx} = {x+y \over x - y}$$
%% CORRECT
{$\tan^{-1}{y\over x} = \ln\sqrt{x^2+y^2} + c$}
%% WRONG
{$(x-y)x^3 = (\ln(x) + c)(x^2 + y^2)$}
%% WRONG
{$\tan^{-1}{y\over x} = \ln(x) + c$}
%% WRONG
{$y^2 = c x^4 - x^2$}
%% WRONG
{$y = c x^2 - x$}


%% PROBLEM
Reduce the following second order equation to a first order equation.
$$y^2 y'' + y(y')^2 = y'$$
%% CORRECT
{${dv\over dy} + {1\over y} v = {1\over y^2}$}
%% WRONG
{$y^2v' + y v^2 = v$}
%% WRONG
{$y^2v'' + y (v')^2 = v'$}
%% WRONG
{$y^2{dv\over dy} + y v^2 = 1$}
%% WRONG
{${dv\over dy} + {1\over y} v^2 = {1\over y^2}$}


%% PROBLEM
Classify the differential equation
$$(x - \cos(y))y' + x\sin(y) = 0$$
%% CORRECT
{{\sl None of the above}}
%% WRONG
{{\sl Linear}}
%% WRONG
{{\sl Separable}}
%% WRONG
{{\sl Exact}}
%% WRONG
{{\sl Homogeneous}}


%% PROBLEM
Solve the initial value problem
$$y'y'' - 6x^2 = 0\qquad y(1) =  y'(1) = 2$$
and evaluate $y(2)$.
%% CORRECT
{$5.73$}
%% WRONG
{$6.85$}
%% WRONG
{$1.23$}
%% WRONG
{$4.08$}
%% WRONG
{$3.41$}


%% END
\end
