{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 3 1}{FONT 0 "-adobe-couri
er-medium-r-normal--14-*" "courier" "Courier" 4 12 192 "Courier-B
old" 12}{FONT 1 "-adobe-courier-bold-o-normal--14-*" "courier" "C
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0 "Helvetica" 10}{SCP_R 1 0 53{COM_R 2 0{TEXT 1 789 "\015\015    \+
      FIRST ORDER EQUATIONS\015          \015             DIRECTI
ON FIELDS\015\015W. E. Boyce\015Rensselaer Polytechnic Institute\
01504/15/94\015\015PURPOSE.  To illustrate the construction of di
rection fields and numerically computed approximations to solutio
n curves for first order equations.  Three examples are worked ou
t and fourteen additional problems are listed which are not solve
d.\015\015\015EXAMPLE 1.  Consider the first order linear equatio
n\015\015   dy/dt = sin(t) - (1/2)y.\015\015(a)  Draw the directi
on field for this equation and use it to describe the behavior of
 solutions.\015\015(b)  Plot several solutions and confirm that t
hey behave in a way consistent with the direction field.\015\015\
015SOLUTION.  Some of the commands that we need reside in the DEt
ools package, so the first step is to bring up this package.\015\
015\015"}}{INP_R 3 0 "> "{TEXT 0 14 "with(DEtools);"}}{OUT_R 4 0 
3{DAG [2,8n4\`DEplot`n4\`DEplot1`n4\`DEplot2`n5\`Dchangevar`n4\`P
DEplot`n5\`dfieldplot`n6\`phaseportrait`}}{SEP_R 5 0}{COM_R 6 0
{TEXT 1 439 "The dfieldplot command will draw the direction field
, but will not plot solutions.  The DEplot command will do both, \+
so we will use it.\015\015Further information about any of these \+
commands can be found online.  For instance, the command ?DEplot \+
brings up a help screen for DEplot.\015\015To use DEplot we must \+
first enter the differential equation.  There are several ways to
 do this, as described in the help screen, one of which is the fo
llowing.\015"}}{INP_R 7 0 "> "{TEXT 0 39 "eq := diff(y(t),t) = si
n(t) - 1/2*y(t);"}}{OUT_R 8 0 7{DAG :3n3\`eq`=3(3n4\`diff`,3(3n3\
`y`,2n3\`t`pD+5(3n3\`sin`pCj2x0001p9/3i2x0001j2x0002}}{SEP_R 9 0}
{COM_R 10 0{TEXT 1 549 "The following DEplot command will produce
 a direction field for this equation.  The first argument is the \+
name of the equation, the second is a list in square brackets of \+
the variables with the independent variable first, the third and \+
fourth are the horizontal and vertical dimensions of the rectangl
e on which the direction field will be drawn.  The arrows option \+
directs Maple to draw a direction field using THIN arrows (it is \+
also possible to request SLIM or THICK arrows).  The title option
 is not essential but can be used to label the plot.\015"}}{INP_R
 11 0 "> "{TEXT 0 62 "DEplot(eq,[t,y],t=0..12,y=-3..3,arrows=THIN
,title=`Figure 1`);"}}{SEP_R 12 0}{COM_R 13 0{TEXT 1 1 "\015"}}
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{COM_R 15 0{TEXT 1 621 "\015\015\015Solutions of this equation ar
e always tangent to the small arrows in the direction field.  The
re seems to be an oscillatory solution in the center of the figur
e, and solutions above or below this curve are moving toward it, \+
or converging.\015\015Note that in drawing a direction field we d
o not SOLVE the differential equation.  Rather, we EVALUATE (repe
atedly) the function on the right side of the equation, which is \+
generally a much easier task.\015\015To plot the solution through
 a given point, we need only to specify the point.  For example, \+
to plot the solution through the origin we modify the DEplot comm
and as follows:\015"}}{INP_R 16 0 "> "{TEXT 0 70 "DEplot(eq,[t,y]
,t=0..12,\{[0,0]\},y=-3..3,arrows=THIN,title=`Figure 2`);"}}
{SEP_R 17 0}{COM_R 18 0{TEXT 1 2 "\015\015"}}{GRH_R 19 0{RASTER 
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0000000000000000000000000000000000000000000}}{COM_R 20 0{TEXT 1 
882 "\015\015Note that the coordinates of the initial point are e
nclosed in brackets, and that the entire list of points (if there
 is more than one) is then enclosed in braces.  Furthermore, the \+
initial point must occur as the fourth argument, that is, between
 the horizontal and vertical dimensions of the rectangle.  If you
 want only the solution and not the direction field, omit the arr
ows option.\015\015If you want Maple to choose the vertical dimen
sion of the rectangle, you can also omit the y specification.  Fo
r a steeply rising or falling curve this may lead to a larger  y \+
 interval than you want.\015\015Sometimes you may want to plot th
e solution through each of several initial points.  One way to do
 this is to list the coordinates of each point in brackets and th
en to enclose the entire collection in braces.  Often it is easie
r to use Maple's seq command to produce the list of points.\015"}
}{INP_R 21 0 "> "{TEXT 0 27 "init := seq([0,i],i=-3..3);"}}{OUT_R
 22 0 21{DAG :3n4\`init`,8[2,3j2x0000i2x0003[2,3p7i2x0002[2,3p7i2
x0001[2,3p7p7[2,3p7j2x0001[2,3p7j2x0002[2,3p7j2x0003}}{SEP_R 23 0
}{COM_R 24 0{TEXT 1 287 "\015In computing a solution curve the de
fault is to use 20 steps to cover the given interval.  In this ca
se this leads to a stepsize of 0.6, which is rather large, and le
ads to a somewhat jagged solution curve.  To produce a smooth cur
ve we use a smaller stepsize in the following command.\015"}}
{INP_R 25 0 "> "{TEXT 0 62 "DEplot(eq,[t,y],t=0..12,\{init\},step
size=0.2,title=`Figure 3`);"}}{SEP_R 26 0}{COM_R 27 0{TEXT 1 2 "\
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00000000000000000}}{COM_R 29 0{TEXT 1 2 "\015\015"}}{COM_R 30 0
{TEXT 1 594 "\015EXAMPLE 2.  Consider the equation\015\015    dy/
dt = 2 - t + ty.\015\015(a)  Draw the direction field for this eq
uation and use it to determine how solutions behave for large t.\
015\015(b)  Plot several solutions to find out if their behavior \+
is consistent with your conclusions from part (a).\015\015(c)  Yo
u should find that for some initial values the corresponding solu
tion tends to + infinity, while for other initial values the solu
tion tends to - infinity.  Estimate, to at least two decimal plac
e accuracy, the initial value that separates these two types of b
ehavior.\015\015\015SOLUTION.   First enter the equation.\015"}}
{INP_R 31 0 "> "{TEXT 0 37 "eq2 := diff(y(t),t) = 2 - t + t*y(t);
"}}{OUT_R 32 0 31{DAG :3n3\`eq2`=3(3n4\`diff`,3(3n3\`y`,2n3\`t`pD
+7j2x0002j2x0001pDi2x0001*5pDp13p9p13p13}}{SEP_R 33 0}{INP_R 34 0
 "> "{TEXT 0 62 "DEplot(eq2,[t,y],t=0..6,y=-3..3,arrows=THIN,titl
e=`Figure 4`);"}}{SEP_R 35 0}{COM_R 36 0{TEXT 1 2 "\015\015"}}
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{COM_R 38 0{TEXT 1 130 "\015\015It appears that solutions above a
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curve tend to minus infinity.\015"}}{INP_R 39 0 "> "{TEXT 0 57 "D
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0000000000000}}{COM_R 43 0{TEXT 1 521 "\015\015\015These solution
s tend to confirm the conclusion reached from the direction field
.  The dividing curve must cross the y axis between -1 and -2.  \
015\015To estimate the critical initial value more accurately, on
e can try initial values in the interval [-2,-1] and gradually br
acket the critical value more closely.\015\015Can you think of an
 alternative way to proceed?\015\015\015\015\015\015\015\015\015E
XAMPLE 3.  Draw a direction field for the equation\015\015  dy/dt
 = - (t/y) - (1/4),\015\015and also plot the solution passing thr
ough the point (0,3).\015\015\015SOLUTION.\015\015"}}{INP_R 44 0 
"> "{TEXT 0 34 "eq3 := diff(y(t),t) = - t/y - 1/4;"}}{OUT_R 45 0 
44{DAG :3n3\`eq3`=3(3n4\`diff`,3(3n3\`y`,2n3\`t`pD+5*5pDj2x0001pA
i2x0001p16/3p16j2x0004p13}}{SEP_R 46 0}{INP_R 47 0 "> "{TEXT 0 46
 "DEplot(eq3,[t,y],t=-4..4,y=-4..4,arrows=THIN);"}}{SEP_R 48 0}
{COM_R 49 0{TEXT 1 216 "\015Based on this direction field, how do
 you think that solutions behave?  In particular, visualize the p
ath followed by the solution starting at the point (0,3).  Then p
lot this solution, using the following command.\015"}}{INP_R 50 0
 "> "{TEXT 0 71 "DEplot(eq3,[t,y],t=-4..4,\{[0,3]\},y=-4..4,arrow
s=THIN,title=`Figure 6`);"}}{SEP_R 51 0}{COM_R 52 0{TEXT 1 2 "\01
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00000}}{COM_R 54 0{TEXT 1 1906 "\015\015Is this plot what you exp
ected to see?  In the central portion of the plot (for small |t|)
 the solution curve closely follows the direction field, but for \+
larger values of |t| the purported solution differs greatly from \+
the direction field.  What has gone wrong?\015\015The explanation
 is that when y = 0 the right side of the differential equation i
s unbounded, corresponding to an infinite slope, or a vertical ta
ngent line.  The interval of definition of the solution ends wher
e the solution reaches the t axis.  The algorithm by which the so
lution is calculated cannot handle infinite slopes (or even extre
mely large slopes) and therefore gives results that are obviously
 incorrect.\015\015This problem is better treated if it is first \+
rewritten as a system of two equations.  That is the subject of a
nother worksheet.\015\015\015\015\015                 PROBLEMS\01
5\015For each of the following differential equations, carry out \+
the following steps.\015\015(a)  Draw the direction field on a su
itable rectangle.\015\015(b)  From your direction field form a co
nclusion as to how the solutions behave, especially as t becomes \+
large.  For instance, you may find it helpful to print a copy of \+
your direction field, and then to sketch by hand several solution
 curves.  If the solutions behave differently, depending on the i
nitial condition that is chosen, estimate the critical initial co
ndition(s) that separate regions of different behavior.\015\015(c
)  Plot a few solution curves to confirm your conclusions from pa
rt (b).\015\015\015 1.  dy/dt = sin(t) + (1/2) y\015\015 2.  dy/d
t = 2 - t - ty\015\015 3.  dy/dt = exp(-t) - 2y\015\015 4.  dy/dt
 = t + exp(-2t) + 3y\015\015 5.  dy/dt = 1 + exp(t) - y\015\015 6
.  dy/dt = sin(t) - y/t\015\015 7.  dy/dt = (1/4) y (3 - y)\015\0
15 8.  dy/dt = (1/4) t y (3 - y)\015\015 9.  dy/dt = 3 t^2/(3y^2 \+
- 4)\015\01510.  dy/dt = (3 t y + y^2)/(t^2 + t y)\015\01511.  dy
/dt = - (3 t y + y^2)/(t^2 + t y)\015\01512.  dy/dt = - sin(2t)/c
os(3y)\015\01513.  dy/dt = sin(ty)\015\01514.  dy/dt = y^2 - t\01
5\015  \015"}}}{END}