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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 265 16 "Direction Fields" }}
{PARA 258 "" 0 "" {TEXT 256 4 "HW 1" }}{PARA 18 "" 0 "" {TEXT 337 9 "S
olutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 62 "Exercise 1  Checking that a Differential Equation is Sati
sfied" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 18 "    \+
              " }}{PARA 0 "" 0 "" {TEXT -1 13 "             " }{TEXT 
329 8 "        " }{XPPEDIT 330 1 "y(x)=A*exp(-1/sqrt(x))*(x^(3/4)+x^(5
/4))+B*exp(1/sqrt(x))*(x^(3/4)-x^(5/4))" "/-%\"yG6#%\"xG,&*(%\"AG\"\"
\"-%$expG6#,$*&F*F*-%%sqrtG6#F&!\"\"F3F*,&)F&*&\"\"$F*\"\"%F3F*)F&*&\"
\"&F*F8F3F*F*F**(%\"BGF*-F,6#*&F*F*-F16#F&F3F*,&)F&*&F7F*F8F3F*)F&*&F;
F*F8F3F3F*F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
328 7 "       " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "is a solution of the differential equation" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "                     \007  " }{TEXT 326 14 "             \+
 " }{XPPEDIT 327 1 "diff( y(x), x,x) = (4+5*x)*y(x)/(16*x^3)" "/-%%dif
fG6%-%\"yG6#%\"xGF)F)*(,&\"\"%\"\"\"*&\"\"&F-F)F-F-F--F'6#F)F-*&\"#;F-
*$F)\"\"$F-!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 179 "Do this by replacing the question marks in the next two execut
ion groups with appropriate arguments. Execute each group by placing t
he cursor within it and pressing the Enter key." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 127 "subs( y(x) = A*exp(-1/sqrt(x))*(x^(3/4)+x^(5/4))+
B*exp(1/sqrt(x))*(x^(3/4)-x^(5/4)) , diff(y(x),x,x) = (4+5*x)*y(x)/(16
*x^3) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-F%6$,&*(%\"AG
\"\"\"-%$expG6#,$*$%\"xG#!\"\"\"\"#F4F,,&*$F2#\"\"$\"\"%F,*$F2#\"\"&F:
F,F,F,*(%\"BGF,-F.6#F1F,,&F7F,F;F4F,F,F2F2,$*(,&F:F,F2F=F,F)F,F2!\"$#F
,\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "simplify( lhs(\")-
rhs(\") );  # Replace the doublequotes \"  with %  in Releases 5 and u
p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 
"Note:  If     " }{TEXT 331 33 "eqn :=  expression1 = expression2" }
{TEXT -1 34 "    then sometimes simplifying    " }{TEXT 332 22 " lhs(e
qn) - rhs(eqn)  " }{TEXT -1 63 "  (to see if  0  results)   is more po
werful than simplifying  " }{TEXT 333 4 " eqn" }{TEXT -1 10 "   itself
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 259 "" 0 "" {TEXT 
-1 45 "Exercise 2  Trajectories and Direction Fields" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 18 "Backgroud Reading:" }
{TEXT -1 58 "  Worksheet: for background on direction field plots     \+
 " }{HYPERLNK 17 "dfieldplotR4.mws" 1 "dfieldplotR4.mws" "" }}{PARA 0 
"" 0 "" {TEXT -1 106 "                                                \+
              for background on working with functions    " }
{HYPERLNK 17 "Tutor1R4.mws" 1 "Tutor1R4.mws" "" }{TEXT -1 93 "        \+
                                                                \nThe \+
help pages for  " }{HYPERLNK 17 "dfieldplot" 2 "DEtools[dfieldplot]" "
" }{TEXT -1 8 "  and   " }{HYPERLNK 17 "display" 2 "plots[display]" "
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 142 "If a 1000 kg bomb is s
ubject only to the force of the Earth's gravity and a linear drag law,
 then the differential equation for its velocity  " }{TEXT 272 1 "v" }
{TEXT 273 1 " " }{TEXT -1 14 " (measured in " }{TEXT 271 3 "m/s" }
{TEXT -1 25 ")  is\n\n                  " }{TEXT 313 12 "            \+
" }{XPPEDIT 314 1 "diff(v(t),t) = -g-.015*v(t)" "/-%%diffG6$-%\"vG6#%
\"tGF),&%\"gG!\"\"*&$\"#:!\"$\"\"\"-F'6#F)F1F," }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 8 "Here    " }{TEXT 275 1 "g" }{TEXT 277 1 " " }{TEXT 276 6 "
= 9.81" }{TEXT 274 87 " m/s/s. (The coefficient 0.015 depends on vario
us factors which are not relevant here.)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 23 "a) Direction Field Plot" }}{PARA 0 "" 0 "" {TEXT -1 71 "Plot th
e direction field of this differential equation in the window   " }
{TEXT 269 26 "[ 0 , 180 ] x [ -600 , 0 ]" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 122 "Do this by replacing the question marks with appr
opriate arguments.  For the last execution group, look up the availabl
e  " }{HYPERLNK 17 "color choices" 2 "color" "" }{TEXT -1 105 "  and p
ick one you like.  (Light colors do not alwayys print out well.)  Exec
ute each line in succession." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "restart:   with(plots):   wi
th(DEtools):  g := 9.81:   # Execute!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "ode := diff(v(t),t) = -g-.015*v(t) ; # Execute!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$odeG/-%%diffG6$-%\"vG6#%\"tGF,,&$!$
\")*!\"#\"\"\"F)$!#:!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 
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" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 164 "Before continuing,  choose a name and assign the plot \+
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{TEXT -1 32 "  find an explicit formula for  " }{TEXT 263 4 "v(t)" }
{TEXT -1 5 ".   \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dsolve
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Define th
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "v1 := unapply( rhs( \" ) , t
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{TEXT -1 25 ".   Use the plot option  " }{TEXT 320 13 "thickness = 2" 
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Before continuing, name t
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If a missile is propelled downwards with initial speed   " }{TEXT 259 
3 "100" }{TEXT 279 1 " " }{TEXT 258 6 "m/s,  " }{TEXT -1 30 "find an e
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " dsolve( \{ ode , v(0) = -100 \} , \+
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                                       The help pages for  " }
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" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "First,  restart the  " 
}{TEXT 295 5 "MAPLE" }{TEXT -1 22 "  kernel and load the " }{TEXT 296 
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" {TEXT -1 51 "a) Using the Direction Field Plot to Spot Solutions" }}
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0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "You should see
 strong evidence that there are a dozen straight line solutions that i
ntersect this viewing window." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Use the   " }{HYPERLNK 17 "subs" 2 "subs
" "" }{TEXT -1 27 "   command to substitute   " }{TEXT 302 14 "y(x) = \+
m*x + b" }{TEXT -1 57 "   into the given equation. Then simplify the e
quation  (" }{HYPERLNK 17 "simplify" 2 "simplify" "" }{TEXT -1 2 ")." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(y(x) = m*x + b, ode);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$,&*&%\"mG\"\"\"%\"xGF*F*%
\"bGF*F+,$-%$sinG6#,(F+!\"\"F(F*F,F*F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "simplify(\");   # Use % instead of \" for releases 5 \+
and up" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"mG,$-%$sinG6#,(%\"xG!\"
\"*&F$\"\"\"F*F-F-%\"bGF-F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 99 "You will see that the left hand side is constan
t. The right hand side can therefore not depend on  " }{TEXT 303 1 "x
" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 36 "What can you conclu
de the value of  " }{TEXT 304 1 "m" }{TEXT -1 5 "  is?" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m := 1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Use  " }{HYPERLNK 17 "solv
e" 2 "solve" "" }{TEXT -1 7 "  or   " }{HYPERLNK 17 "fsolve" 2 "fsolve
" "" }{TEXT -1 80 "  (with the three parameters: equation, variable, r
ange)  to find the value of  " }{TEXT 305 1 "b" }{TEXT -1 25 "  that i
s least negative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "fsolve( -arc
sin(1) = b, b, -2 .. 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+Fjzq:!
\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve( 1 = - sin(b),
 b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#!\"\"\"\"#" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "b) The General So
lution" }}{PARA 0 "" 0 "" {TEXT -1 4 "Use " }{TEXT 306 5 "MAPLE" }
{TEXT -1 59 "   to find a general solution to the differential equatio
n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dsolve(ode, y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/,&*$,&!\"\"\"\"\"-%$tanG6#,&%\"xG#F(\"\"#-%\"yG6#F-#F'F/F(F'!\"#F-F'
%$_C1G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
20 "The solution  that  " }{TEXT 299 5 "MAPLE" }{TEXT -1 46 "  returns
 might be implicit . If so then use  " }{MPLTEXT 1 0 5 "solve" }{TEXT 
-1 16 "  to solve for  " }{MPLTEXT 1 0 4 "y(x)" }{TEXT -1 13 "  explic
itly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "eqn := y(x) = solve(\" ,
 y(x));   # This line may be unnecessary with some Maple releases." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"yG6#%\"xG,&F)\"\"\"-%'arct
anG6#*&,(!\"#F+F)F+%$_C1GF+F+,&F)F+F2F+!\"\"F1" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "One particular solution o
f the differential equation is  " }{MPLTEXT 1 0 15 "y(x) = x - Pi/2" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "There is no  value of th
e arbitrary constant   " }{MPLTEXT 1 0 3 "_C1" }{TEXT -1 75 "   that c
orresponds to this solution (or any other straight line solution)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "However, \+
plot the solutions that correspond to " }{MPLTEXT 1 0 8 "_C1 = 10" }
{TEXT -1 8 "  and   " }{MPLTEXT 1 0 9 "_C1 = 100" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(subs(_C1 = 10, rhs(eqn)
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "These plo
ts should give you a clue as to how to recover the particular solution
  " }{MPLTEXT 1 0 15 "y(x) = x - Pi/2" }{TEXT -1 37 "  from the genera
l solution given by " }{MPLTEXT 1 0 3 "eqn" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 24 "Namely, calculate the   " }{HYPERLNK 17 "limit" 
2 "limit" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 30 "                              " }{TEXT 309 10 "          " }
{XPPEDIT 310 1 "limit(x-2*arctan((-2+x+_C1)/(x+_C1)),_C1=infinity)" "-
%&limitG6$,&%\"xG\"\"\"*&\"\"#F'-%'arctanG6#*&,(F)!\"\"F&F'%$_C1GF'F',
&F&F'F0F'F/F'F//F0%)infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y(x) = limit(x-2*arctan((-2+
x+_C1)/(x+_C1)),_C1 = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%\"yG6#%\"xG,&F'\"\"\"%#PiG#!\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Exercise 4  The  " }{TEXT 294 5 "
MAPLE" }{TEXT -1 12 "  Function  " }{TEXT 293 6 "DEplot" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 18 "Backgroud Reading:
" }{TEXT -1 21 "  The help page for  " }{HYPERLNK 17 "DEplot" 2 "DEtoo
ls[DEplot]" "" }{TEXT -1 105 "  is rather involved. The next execution
 group provides you with an example of how to use the function.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 335 7 "Caveat:" 
}{TEXT -1 3 "   " }{TEXT 334 6 "DEplot" }{TEXT -1 3 "   " }{TEXT 336 
237 "uses a default number of steps.  If you use too great an interval
 then the resulting step size may be too large. If your plot suggests \+
that this is happening, then go to the linked help page for DEplot to \+
see how to adjust the step size." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 69 "Look at the commands in the next executio
n group to see how we use   " }{HYPERLNK 17 "DEplot" 2 "DEplot" "" }
{TEXT -1 50 "   to plot a solution to the Initial Value Problem" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "\022 " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "    " }
{TEXT 280 44 " y'(x) = sin(x^2/(1+x^2)+y(x)),   y(-5) = 6:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 54 "PLease read and then execute the next execution group.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "restart:  with(DEtools): wi
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 b := 12:\nDEplot(ode , [y(x)] , x = -5 .. 12, [[ y(-5)=6 ]] , stepsiz
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