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{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam II}\quad{\sl March 22, 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 6 points each and 6 True/False question worth 3 points each.
\bigskip\bigskip

{
  \baselineskip=20pt
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  \answersheet{1}{14}{1}{2}{\theanswers{}{}{}{}{}}
  \hrule
  \bigskip
  \answersheet{15}{20}{1}{2}{{\it (T)\qquad (F)}}
  \hrule
}

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\hfill Score \hbox to 1truein{\hrulefill}

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

\newproblem
Solve the differential equation
$$y' + y = 6 e^x$$
subject to the initial condition $y(1) = 1$.
\answers
{$y = 3e^{x}$} 
{$y = 3e^x - 3 e^{-x} + 1$}
{$y = 3e^{1-x} - 3 e^{x-1} + 1$}
{$y = e^{x-1} - 3e^{x} + 3e$}
{$y = 3e^x - 3 e^{2-x} + e^{1-x}$}

\newproblem
On what interval is the solution to the differential equation
$$ y' - \tan(x) y = e^x,\qquad y(\pi) = 0$$
guaranteed to exist?
\answers
{$(-{\pi\over2},{\pi\over2})$}
{$(0,2\pi)$}
{$(-2\pi,2\pi)$} 
{$({\pi\over2},{3\pi\over2})$}
{$(-\infty,\infty)$}

\newproblem
Consider the differential equation
$$(x^2 - y)^2 y' = e^x$$
Determine the region in the $xy$-plane where the existence of a unique
solution through any specified point is guaranteed to exist.
\answers
{$\{(x,y)\,|\,y \ne x^2\}$}
{$\{(x,y)\,|\,y \ne \pm x\}$}
{{\sl all $x$ and $y$}\endanswerline}
{$\{(x,y)\,|\,y \ne 0, x\ne 0\}$} 
{$\{(x,y)\,|\,(y^2 - x)^2\ne e^x\}$}


\newproblem
Solve the differential equation
$$y' = {x y^2 + y \over (x^2 y + x)}$$
\answers
{$y = c x$}
{$y = c {1\over x}$}
{${1\over3}x^3y + xy^2 + {1\over2}y^2 = c$\endanswerline}
{${1\over2}x^2y^2 + xy^2 + {1\over2}y^2 = c$} 
{$y ={1\over2}x^2y^2 + c{1\over x}$}


\newproblem
A 1000~gal tank contains 500~gal of water having a 2\% concentration of
impurities. Pure water is added to the tank at a rate of 5~gal/min. At the
same time the water in the tank is being filtered. The filtering process
takes 10~gal/min from the tank, removes 50\% of the impurities, and returns
the filtered water to the tank at the same rate. The tank is kept
well mixed. What is the concentration of impurities when the tank is filled
to capacity?
\answers
{0.1\%}
{0.2\%} 
{0.5\%}
{1.0\%}
{5.0\%}



\newproblem
Suppose a certain sum is deposited in a bank that pays interest at an annual
rate of~$r$ compounded continuously. Find the time~$T$ required for the
original sum to double in value as a function of the interest rate~$r$.
\answers
{$T = \log_2(r)$} 
{$T = {\ln(r) \over \ln(2)}$}
{$T ={\ln(2) \over \ln(r)}$}
{$T = {\ln(2)\over r}$}
{$T = {r \over \ln(2)}$}


\newproblem
What is the limiting value of a population $N(t)$ if $N(0) = 10$ and $N$
satisfies the logistic equation
$${dN\over dt} =N(N^2 - 400)$$
\answers
{$-\infty$}
{$-20$}
{$0$}
{$20$}
{$\infty$} 


\newproblem
Find the {\sl stable} equilibrium solutions to the differential equation
$${dx\over dt} = 5(0.5 - x)(0.3 - x)(0.2 - x)$$
\answers
{$x = 0.3$}
{$x = 0.2$ and $x = 0.5$}
{$x = 0.2$, $x = 0.3$, and $x = 0.5$\endanswerline}
{$x = 0$} 
{{\sl There are no stable equilibria}}


\newproblem
A body of constant mass is launched vertically upward from sea level with an
initial velocity of 5000~mi/hr. Neglecting air resistance, but considering
the change of gravitational attraction with altitude,~$x$, find the maximum
altitude attained by the body. The appropriate differential
equation is
$$v {dv\over dx} = -{gR^2\over (R+x)^2}$$
Assume $R = 4000{\rm\ mi}$ and $g = 32{\rm\ ft/sec^2} = 78545{\rm\ mi/hr^2}$.
\answers
{149.9 mi}
{153.4 mi} 
{159.1 mi}
{165.7 mi}
{167.3 mi}

\newproblem
Find the integral curve to the differential equation
$$(e^{xy} y + 3 x^2 y - y^2) dx + (e^{xy} x + x^3  - 2 x y - 1) dy = 0$$
which passes through the point $(0,2)$.
\answers
{$y = 1$}
{$e^{xy} + xy(x^2 - y) - y = 0$}
{$f(x,y) = e^{xy} + xy(x^2 - y) - y$\endanswerline}
{$e^{xy} + xy(x^2 - y) + 1 = 0$} 
{$e^{xy} + xy(x^2 - y) - y + 1 = 0$}

\newproblem
Determine an integrating factor $\mu$ for the differential equation
$$\tan(y) \, dx + (x + y \tan(y) - 1)\, d y = 0$$
\answers
{$\mu = \tan(y) \ln(x+y \tan(y) - 1)$}
{$\mu = {1\over x}$\endanswerline} 
{$\mu = x\sec^2(y) - x$}
{$\mu = \sin(y)$}
{$\mu = \cos(y)$}

\newproblem
Let $\phi(x)$ be the solution to the differential equation
$$y' = {y^2 + x y - x^2\over x^2}$$
satisfying $\phi(1) = 0$. Find $\phi(2)$.
\answers
{$3\over\displaystyle4$}
{$2\over\displaystyle5$}
{$-{1\over\displaystyle3}$}
{$-{2\over\displaystyle3}$}
{$-{6\over\displaystyle5}$}

\newproblem
Classify the following differential equation
$$x y' - 2 y = \sqrt{x^2 - y^2}$$
\answers
{Linear}
{Separable}
{Exact}
{Homogeneous} 
{None of the above}


\newproblem
Reduce the following differential equation to a first order equation.
$$y'' + y(y')^3 = 0$$
\answers
{${dv\over dy} = -v^3$}
{${dv\over dy} = -yv^2$}
{$y' +{1\over2}y^2(y')^3 = 0$}
{$v' + yv^3 = 0$\endanswerline}
{$v' + yv^3 = -v$} 
\hrule
\bigskip\bigskip
{\bf True/False}
\bigskip\bigskip
\newproblem
If $p(x)$ and $g(x)$ are continuous in the interval $(\alpha, \beta)$ then a
solution to the equation $y' + p(x)y = g(x)$ exists and is valid on the
interval $(\alpha, \beta)$.
\endproblem

\newproblem
If $y_1$ and $y_2$ are solutions to the differential equation
$y'' + p(x)y' + q(x) y = 0$, then $c_1y_1 + c_2y_2$ is also a solution of
the equation.
\endproblem

\newproblem
If $y_1$ and $y_2$ are solutions to the differential equation
$y'' + p(x)y' + q(x) y = 0$, then any solution can be written as $c_1y_1 +
c_2y_2$ for some choice of constants $c_1$ and $c_2$.
\endproblem

\newproblem
If $y_1$ and $y_2$ are solutions to the differential equation
$y''= f(x,y,y')$, then $c_1y_1 + c_2y_2$ is also a solution of
the equation for some choice of constants $c_1$ and $c_2$.
\endproblem

\newproblem
The functions $y_1 =\sqrt{x}$ and $y_2 = {1\over\sqrt{x}}$ are a fundamental
set of solutions for the equation $xy'' + y' -{1\over4x}y = 0$,~$x > 0$.
\endproblem

\newproblem
There is a second order linear homogeneous differential equation such that the
functions $y_1 = x^2$ and $y_2 = x^3$ form a fundamental set of solutions for
all~$x$. (Hint: check the Wronskian.)
\endproblem


\bye