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\topmatter

{\bf Name:}  \vrule height .4pt depth 0 pt width 2.0
in  \hskip .5 in {\bf Professor:} A.Alexandrou Himonas
\bigskip
\bigskip
\bigskip

 \centerline{{\bf {Mathematics 226}}
\centerline{{\bf {Differential Equations and Linear Algebra}}
\centerline{{\bf Fall Semester 1997}}
\centerline{{\bf Exam 1}}
\centerline{{\bf {September 23, 1996}}



\bigskip
\bigskip
\bigskip
\hrule
\bigskip
This Examination contains five problems worth a total of  100  points,
each problem worth 20 points, on (7) sheets of paper including the
front cover and one extra sheet on the back.  Do all your work in this
booklet and show your computations.  Calculators, books and notes are
not allowed.
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\hrule
\bigskip
\bigskip

$$\vbox{\offinterlineskip
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\ \  1 & \qquad \qquad \cr
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\bigskip
\bigskip

{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Test}":
\bigskip
\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
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\centerline{\bf GOOD LUCK}


\newpage
\noindent
{\bf 1.}  Find the general solution of the differential equations:
$$x\frac{dy}{dx}+(y+1)^2=0\tag1.1$$
\vskip 2.7 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in
\bigskip
$$x\frac{dy}{dx}=y-xe^{y/x}\tag 1.2$$
\vskip 2.6 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in

\newpage
\noindent
{\bf 2.}  Solve the following initial value problem:
$$(1+x^4)\frac{dy}{dx}+4x^3y=1,\ y(0)=7.$$

\vskip 6 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in

\newpage
\noindent
{\bf 3.} Find $a$ for which the following differential equation
 is exact and then solve it.
$$(axy+x^2+3)+(x^2+y^3+2y)\frac{dy}{dx}=0$$

\vskip 6 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in

\newpage
\noindent
{\bf 4a.} A skydiver weighing 192 lbs.( including equipment) falls vertically
downward from a certain high altitude.  The parachute opens
 when the skydiver reaches speed equal to 160 ft/sec.
Assume that $ t = 0 $ when the parachute opens and that the air resistance
is $ 12|v|$. Also assume that $g=32\, ft/sec^2$.
 \medskip
\noindent
{\bf i)}  Write the initial value problem for the speed $ v(t) $ at any time $
t.${\bf DO NOT SOLVE IT!}
\smallskip
\noindent
{\bf ii)}  Find the limiting velocity $ v_L $ of the skydiver after the
parachute opens.
\medskip
\noindent
\vskip 1.5 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in
\bigskip
\noindent
{\bf 4b.} Find the constant (equilibrium) solutions of the differential
equation
$$\frac{dy}{dt}=-0.01(1-\frac y2)(1-\frac y8)y.$$
Then classify each one  as asymptotically stable, or unstable,
and  sketch their graphs. Finally sketch  the graph of the solution $y=y(t)$
with initial value $y(0)=3$ (indicate its behavior for large $t$).

\vskip 3.3 in

\newpage
\noindent
{\bf 5a.}  Consider a tank holding initially 100  gallons of a salt
solution with
concentration  0.2 lb of salt per gallon.  A solution containing  0.4
lb of salt per gallon is pumped into the tank at the rate of  5  gal/min,
and the
well-stirred mixture flows out of the tank at the rate of  2  gal/min.
	Write the initial value problem (i.e. a  differential equation
  and an initial condition)
needed to find the amount  $S(t)$ of salt in the tank at any time  $t$.
{\bf DO NOT SOLVE.}

\vskip 2.5 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in
\bigskip
\noindent
{\bf 5b.} Circle the differential equation whose direction fields
are shown in the following picture.
$$A.\  y'=y(y-x),\quad B.\  y'=(y-1)x,\quad C.\  y'=(y-1)(x-y),\quad
D.\  y'=1-x^2,\quad E.\  y'=\sin x$$
$$\epsfxsize=3.0in\epsfbox{dir.fields.exam.1}$$

\newpage
\noindent
.










\bye













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\documentstyle{amsppt}
\magnification 1200
\NoRunningHeads
\pageno=1
\topmatter

{\bf Name:}  \vrule height .4pt depth 0 pt width 2.0
in  \hskip .5 in {\bf Professor:} A.Alexandrou Himonas
\bigskip
\bigskip
\bigskip

 \centerline{\bf {Mathematics 226}}
\centerline{\bf {Differential Equations and Linear Algebra}}
\centerline{\bf Fall Semester 1996}
\centerline{\bf Exam 1}
\centerline{\bf {September 23, 1996}}

\bigskip
\bigskip
\bigskip
\hrule
\bigskip
This Examination contains five problems worth a total of  100  points,
each problem worth 20 points, on (7) sheets of paper including the
front cover and one extra sheet on the back.  Do all your work in this
booklet and show your computations.  Calculators, books and notes are
not allowed.
\bigskip
\hrule
\bigskip
\bigskip

$$\vbox{\offinterlineskip
\halign{
\vrule  height .4 in depth 0 in width .4pt# &
\vrule  height .4 in depth 0 in width .4pt#
\vrule  height .4 in depth 0 in width .4pt \cr
\noalign{\hrule}
\ \  1 & \qquad \qquad \cr
\noalign{\hrule}
\ \  2  & \qquad \qquad \cr
\noalign{\hrule}
\ \ 3 & \qquad \qquad \cr
\noalign{ \hrule}
\ \  4 & \qquad \qquad \cr
\noalign{ \hrule}
\ \  5  & \qquad\qquad  \cr
\noalign{\hrule}
\ Total & \qquad \qquad \cr
\noalign{\hrule}
}}$$
\bigskip
\bigskip

{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Test}":
\bigskip
\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\bigskip
\centerline{\bf GOOD LUCK}


\newpage
\noindent
{\bf 1.}  Find the general solution of the differential equations:
$$x\frac{dy}{dx}+(y+1)^2=0\tag1.1$$
\vskip 2.7 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in
\bigskip
$$x\frac{dy}{dx}=y-xe^{y/x}\tag 1.2$$
\vskip 2.6 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in

\newpage
\noindent
{\bf 2.}  Solve the following initial value problem:
$$(1+x^4)\frac{dy}{dx}+4x^3y=1,\ y(0)=7.$$

\vskip 6 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in

\newpage
\noindent
{\bf 3.} Find $a$ for which the following differential equation
 is exact and then solve it.
$$(axy+x^2+3)+(x^2+y^3+2y)\frac{dy}{dx}=0$$

\vskip 6 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in

\newpage
\noindent
{\bf 4a.} A skydiver weighing 192 lbs.( including equipment) falls vertically
downward from a certain high altitude.  The parachute opens
 when the skydiver reaches speed equal to 160 ft/sec.
Assume that $ t = 0 $ when the parachute opens and that the air resistance
is $ 12|v|$. Also assume that $g=32\, ft/sec^2$.
 \medskip
\noindent
{\bf i)}  Write the initial value problem for the speed $ v(t) $ at any time $
t.${\bf DO NOT SOLVE IT!}
\smallskip
\noindent
{\bf ii)}  Find the limiting velocity $ v_L $ of the skydiver after the
parachute opens.
\medskip
\noindent
\vskip 1.5 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in
\bigskip
\noindent
{\bf 4b.} Find the constant (equilibrium) solutions of the differential
equation
$$\frac{dy}{dt}=-0.01(1-\frac y2)(1-\frac y8)y.$$
Then classify each one  as asymptotically stable, or unstable,
and  sketch their graphs. Finally sketch  the graph of the solution $y=y(t)$
with initial value $y(0)=3$ (indicate its behavior for large $t$).

\vskip 3.3 in

\newpage
\noindent
{\bf 5a.}  Consider a tank holding initially 100  gallons of a salt
solution with
concentration  0.2 lb of salt per gallon.  A solution containing  0.4
lb of salt per gallon is pumped into the tank at the rate of  5  gal/min,
and the
well-stirred mixture flows out of the tank at the rate of  2  gal/min.
	Write the initial value problem (i.e. a  differential equation
  and an initial condition)
needed to find the amount  $S(t)$ of salt in the tank at any time  $t$.
{\bf DO NOT SOLVE.}

\vskip 2.5 in
\hskip 2.0 in {\bf Answer:}  \vrule height .4pt depth 0 pt width 2 in
\bigskip
\noindent
{\bf 5b.} Circle the differential equation whose direction fields
are shown in the following picture.
$$A.\  y'=y(y-x),\quad B.\  y'=(y-1)x,\quad C.\  y'=(y-1)(x-y),\quad
D.\  y'=1-x^2,\quad E.\  y'=\sin x$$
$$\epsfxsize=3.0in\epsfbox{dir.fields.exam.1}$$

\newpage
\noindent
.










\bye
































