%The following file (in AMS-TeX) contains the first exam for math 231 when
%I taught it in 1994. \frac {A}{B} or \frac AB means A/B.

\input amstex
\documentstyle{amsppt}
\magnification=1200
\NoBlackBoxes
\document

\topmatter
\title
Differential equations-1st. exam- September 27,1994
\endtitle
\endtopmatter

{\bf 1.}[25]Solve the initial value problem:
$$xy'+y=e^x,\quad y(1)=1.
$$
Include the interval where the solution is defined. Remember to
reduce to `standard form' first!

\vskip .5cm

{\bf 2.}
[20]A population model is described by the differential equation:
$$\frac{dp}{dt}=-(1-\frac p2)(1-\frac p3)p.
$$
($p$ in units of $10^3$ individuals, $t$ in years.) 
\item{(i)}[7] Sketch the `flow diagram' for the model;
\item{(ii)}[7] What is the asymptotic behavior of a population
whose initial number of individuals is 2500?
\item{(iii)}[6] Are all solutions of the model guaranteed to exist for
all $t>0$? Explain.
\vskip .5cm

{\bf 3.}
[25]Consider the equation:
$$
\frac{dy}{dx}=-\frac{2x+y}{x+3y}.
$$
\item{(i)}[10] Show the equation is exact;
\item{(ii)}[10] Find the solution of the initial value problem $y(1)=1$
in implicit form.
\vskip .5cm

{\bf 4.} [20]A swimming pool whose volume is 10,000 gal. contains water that is
0.01\% chlorine. At $t=0$, city water with 0.001\% chlorine is pumped
into the pool at 5 gal/min; the water in the pool flows out at the same rate. 
What is the concentration of chlorine in the pool after one hour? When
will the concentration of chlorine in the pool equal 0.002\% ?
\vskip .5 cm

{\bf 5.}[10]A building with no heating or cooling has a time constant of
2 h. Assuming the outside temperature varies as a sine wave with
an average value of 60 F reached at 8 a.m. and a maximum of 80 F at 2 p.m.,
write down the differential equation describing the temperature
inside the building as a function of time. (You do not
have to solve the equation). Take the time origin ($t=0$) to correspond
to 8 a.m. , and the period of the sine wave to be 24h.

\enddocument

*****************************************************************************
The file below (also in TeX) contains the second exam for my Math231
class in 1994.
****************************************************************************

\input amstex
\documentstyle{amsppt}
\magnification=1200
\NoBlackBoxes

\document

\head
MATH 231-SECOND EXAM-November 1,1994
\endhead

{\bf 1.}Find the general solution:
$$y''-3y'+2y=e^t\sin t.$$

\vskip .5cm

{\bf 2.}Find a general solution to the equation below.
$$(x+1)^2y''+10(x+1)y'+14y=0,\quad x>-1.$$

\vskip .5cm

{\bf 3.}A mass of .5 kg stretches a spring vertically by 10cm
in its equilibrium position. The mass is acted upon by an
external force of $10\sin t$ N and moves vertically in a medium that
imparts a viscous force of 2N when the speed of the mass is 4cm/s.

(a)If the mass is set in motion from its equilibrium position
with an initial velocity of 8cm/s, formulate the initial value
problem describing its motion. (Use $g=10m/s^2$ if necessary).

(b)Find the steady-state solution of the equation in (a).

(c)Assume now that no viscous force acts on the mass. What is
the resonance frequency of the system?

\vskip .5cm
{\bf 4.}Use the method of reduction of order to find the general
solution of:
$$(x-1)y''-xy'+y=0,\quad x>1,$$
given that $y_1(x)=e^x$ is a solution.

\enddocument