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{SECT 0 {EXCHG {PARA 256 "" 1 "" {TEXT -1 37 "M2280 - 1      FIRST MAP
LE ASSIGNMENT" }}{PARA 0 "" 0 "" {TEXT -1 44 "Treibergs               \+
 Due Sept. 10, 2003." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 202 "In the first assignment, we shall explore some basic M
APLE capabilities in solving differential equations. If you are unfami
liar with MAPLE, you should work through Prof. Korevaar's MAPLE introd
uction (" }{HYPERLNK 17 "Korevaar's Tutorial as MAPLE worksheet " 1 "M
2250falltut.mws" "" }{TEXT 281 62 "(http://www.math.utah.edu/~korevaar
/2250fall03/2250falltut.mws" }{TEXT -1 5 ") or " }{TEXT 280 2 "  " }
{TEXT -1 1 " " }{URLLINK 17 "Korevaar's tutorial (PDF)" 4 "http://www.
math.utah.edu/~korevaar/2250fall03/2250falltut.pdf" "" }{TEXT -1 2 " (
" }{TEXT 282 61 "http://www.math.utah.edu/~korevaar/2250fall03/2250fal
ltut.pdf" }{TEXT -1 300 ")  The page numbers correspond to a different
 text than ours.) He develops enough tools for doing the second exerci
se, about the variation of the inside temperature of an unheated Salt \+
Lake City apartment due to outside temperature variations. This follow
s the Computing Project 1.5 from the text by " }{TEXT 256 18 "Edwards \+
and Penney" }{TEXT -1 194 " (p.56 in 3rd ed.) \n\nPlease work the prob
lems at the bottom of this page. These are different questions than fo
r M2250, so you will not be able to use Korevaar's templates without s
ome revision." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 170 "We start by solving differential equations, following Co
mputer Project 1.6. (p. 73 in 3rd ed.) We illustrate some basic MAPLE \+
constructs. Suppose we wished to check that\n" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "   " }{TEXT 257 65 "y(x)= 5e^(3x) - 6e^(-4x) - x^3/12 - x^
2/48 - 13 x^2/288 -25/3456\n" }}{PARA 0 "" 0 "" {TEXT -1 35 "satisfies
 the differential equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 3 "   " }{TEXT 258 29 " y'' + y' - 12 y - x^3  =  0.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Enter
 a function  " }{TEXT 259 6 "y(x). " }{TEXT -1 71 "Recall that := is t
he assignment operator, thus the function is named  " }{TEXT 260 3 "y.
\n" }{TEXT -1 50 "It is always a good idea to clear your variables. " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y:= x->5*
exp(3*x)-6*exp(-4*x)-x^3/12-x^2/48-13*x/288-25/3456;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,.*&\"\"&\"
\"\"-%$expG6#,$*&\"\"$F/9$F/F/F/F/*&\"\"'F/-F16#,$*&\"\"%F/F6F/!\"\"F/
F>*&#F/\"#7F/*$)F6F5F/F/F>*&#F/\"#[F/*$)F6\"\"#F/F/F>*&#\"#8\"$)GF/F6F
/F>#\"#D\"%cMF>F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The valu
e of the function along with the first derivative at zero is" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y(0);D(y)(0);" }{TEXT -1 0 "
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!%\"[$\"%cM" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"&>7\"\"$)G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "W
e form the differential expression, where  " }{TEXT 261 14 "diff(y(x),
x,x)" }{TEXT -1 37 " means differentiate the expression  " }{TEXT 262 
4 "y(x)" }{TEXT -1 18 "  with respect to " }{TEXT 263 1 "x" }{TEXT -1 
7 " twice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "diff(y(x),x,x
)+diff(y(x),x) - 12*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"xG
\"\"$\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The result is  x^3
, which is the desired one. We'll see another way to check later, usin
g   " }{TEXT 264 8 "odetest." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Suppose that we wished to show \+
that  " }{TEXT 265 11 "z(x)= x e^x" }{TEXT -1 13 "  satisfies  " }
{TEXT 266 26 "x^2 z'' - x(x+1)z' + z = 0" }{TEXT -1 3 "  ," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "z:= x->x*exp(x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"zGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%
$expG6#F-F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x^2*di
ff(z(x),x,x)-x*(x+1)*diff(z(x),x)+z(x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(*&)%\"xG\"\"#\"\"\",&*&F'F(-%$expG6#F&F(F(*&F&F(F+F(F(F(F(*(F&
F(,&F&F(F(F(F(,&F+F(F.F(F(!\"\"F.F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 263 "This is too complicated. MAPLE can simplify expressions in a n
umber of ways. One way to try is to call the operation simplify(% ). %
  is MAPLE's way of referring to the previously displayed experession.
 Simplification yields zero, so the expression is a solution." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Ther
e are strong ODE specific tools. First the ODE library is loaded. Then
 a variable " }{TEXT 267 3 "eq1" }{TEXT -1 74 " is stored with a diffe
rential equation involving the unknown expression  " }{TEXT 268 4 "u(x
)" }{TEXT -1 32 ".  I did not call the function  " }{TEXT 269 4 "y(x)
" }{TEXT -1 57 " because this already has a different meaning for MAPL
E. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eq1:= diff(u(x),x,x)+diff(u(
x),x) - 12*u(x)-x^3=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,*-%
%diffG6$-%\"uG6#%\"xG-%\"$G6$F-\"\"#\"\"\"-F(6$F*F-F2*&\"#7F2F*F2!\"\"
*$)F-\"\"$F2F7\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Now we se
e some of MAPLE'S power. Invoking the DE solver we look for the genera
l solution of the ODE." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "d
solve(eq1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"uG6#%\"xG,.*&-%$ex
pG6#,$*&\"\"%\"\"\"F'F0!\"\"F0%$_C2GF0F0*&-F+6#,$*&\"\"$F0F'F0F0F0%$_C
1GF0F0#\"#D\"%cMF1*(\"#8F0\"$)GF1F'F0F1*&\"#[F1F'\"\"#F1*&\"#7F1F'F8F1
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "WOW! This is the general sol
ution, where _C1 and _C2 are constants of integration. If we wish to a
dd initial conditions, we can.  Trying the initial conditions from bef
ore," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sol1:=dsolve(\{eq1,
u(0)=-3481/3456, D(u)(0)=11219/288\});" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%sol1G/-%\"uG6#%\"xG,.*&\"\"&\"\"\"-%$expG6#,$*&\"\"$F-F)F-F-F
-F-*&\"\"'F--F/6#,$*&\"\"%F-F)F-!\"\"F-F;*&\"#7F;F)F3F;*&\"#[F;F)\"\"#
F;*(\"#8F-\"$)GF;F)F-F;#\"#D\"%cMF;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 193 "Thus MAPLE finds the solution from the ODE and IC. DEtools als
o has a routine that plugs functions into differential equations, whic
h we did above in a more bare-hands way. It is the procedure " }{TEXT 
270 9 "odetest. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "odetest
(sol1,eq1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 13 "From page 71," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "dsolve(D(v)(x)-sin(x - v(x)),v(x));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"vG6#%\"xG,&F'\"\"\"*&\"\"#F)-%'arctanG6#*&,(F'F
)F+!\"\"%$_C1GF1F),&F'F)F2F1F1F)F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 63 "Here is an illustration of slope fields. this is problem 28[24]
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eq3:= D(w)(x)=x+w(x)^2/
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/--%\"DG6#%\"wG6#%\"xG,&F
,\"\"\"*&#F.\"\"#F.*$)-F*F+F1F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 13 "You can try :" }{TEXT 279 23 " dsolve( eq3, w(0)=1); " }{TEXT 
-1 140 "But the answer is expressed in certain advanced special functi
ons.\nIt is more meaningful to plot the slope fields with several traj
ectories." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "DEplot( eq3, \+
w(x), x=-1..0.5, \{seq([w(0)=k/2],k=-2..4)\}, color=black, linecolor=[
red,magenta,blue,green,cyan,coral,navy],arrows=LINE);" }}{PARA 13 "" 
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b\\l$\"3U2=oerD%\\\"F*7$F_e^l7$Feb\\l$\"3eG2b3z2a:F*7$Fce^l7$F[c\\l$\"
3*Rnh$[QFF;F*7$Fge^l7$Fac\\l$\"3A'\\c7Htfr\"F*7$F[f^l7$Fgc\\l$\"3+OEo#
e&)G#=F*7$F_f^l7$F\\d\\l$\"3T!)pJip_^>F*7$Fcf^l7$Fbd\\l$\"39iF-tvb1@F*
7$Fgf^l7$Fid\\l$\"3/R$f0&zH%H#F*7$F[g^l7$F_e\\l$\"3kZ4_')=bBDF*7$F_g^l
7$Fee\\l$\"3s*o/3$['p!GF*7$Fcg^l7$F[f\\l$\"3/@%yi,\"QjJF*7$Fgg^l7$Faf
\\l$\"3l3^tcL<AOF*7$F[h^l7$Fgf\\l$\"3Ypfy;myJUF*-Ff^\\l6&Fh^\\l$\")!\\
DP\"F^g\\lFdh^l$\")viobF^g\\lF`g\\lFdg\\l-%%VIEWG6$;$!++++v5F`_\\l$\"+
+++]dFf_\\l;$!+_)3OZ\"F`_\\l$\"+;@Z.XF`_\\l-%+AXESLABELSG6$Q\"x6\"Q%w(
x)Fii^l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curv
e 8" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 37 "M2280 - 1  FIRST MAPLE LAB QUESTIONS." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 332 "Be sure your nam
e and student number are on your paper. Do not hand in your working th
rough my example problems. Write up solutions in a self contained mann
er. Use enough comments to explain your analysis. Have MAPLE produce t
he solutions and graphs. label these by hand if necessary. Hand in a p
rinted hard copy of your solutions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 107 "1. Use MAPLE to solve any one of the p
roblems from section 1.6 page 71 (all page numbers from text 3rd ed.)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 766 "2.  \+
(Taken from Korevaar's M2250 fall 2003) You are to do the section 1.5 \+
computer project. A preliminary discussion, in which the text discussi
on (pages 56-58, 3rd ed.) is expanded to include Maple commands, is fo
und at the end of Prof. Korevaar's tutorial which accompanies this pro
ject.  It is assumed that you have already worked through that.  \n\nS
CENARIO:\nIt is not summer in Georgia.  It is March in Salt Lake City.
  We will dream it is early winter and that the 5-day forecast in the \+
newspaper says the weather will be stable, with lows of 15 degrees,  a
nd highs of 55 degrees.   As in the text and for the sake of simplicit
y, we will assume a sinusoidal daily temperature oscillation, except o
ur low will be at 2:30 a.m, and our  high will be at 2:30 p.m.  \n" }}
{PARA 0 "" 0 "" {TEXT -1 201 "We must leave town for 3 days, and are d
eciding whether to turn off the heat while we are gone.  The question \+
we wonder about is, will the water pipes in the house freeze if we do \+
turn off the heat?   " }}{PARA 0 "" 0 "" {TEXT -1 571 "\na. Review the
 model which leads to equation (3) on page 57.  Use integrals #49 and \+
#50 (flyleaf) and the algorithm for solving first order linear DE's in
 section 1.5 (p. 47) to solve (3) by hand, keeping all parameters as l
etters.   Staple your work onto the printout of your completed project
 which you hand in.  Your answer should agree with equation (4) on pag
e 57.  You will see that the constant c0 in that answer is the constan
t C of integration you obtain when you follow the solution recipe for \+
linear DE's.  You find its value in terms of the initial condition " }
{TEXT 276 7 "u(0)=u0" }{TEXT -1 15 " by pluggin in " }{TEXT 277 9 "u=u
0, t=0" }{TEXT -1 12 ", as usual. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 312 "b.  Use dsolve to have Maple find the so
lution to (3), with u(0)=u0. You might have already done this in the t
utorial, in which case you can copy the appropriate commands  from the
re, paste them in here, and re-execute them.  Check that your solution
 agrees with your hand work above, as well as with the text's." }}
{PARA 0 "" 0 "" {TEXT -1 123 "\nc.  Figure out the parameter values fo
r the Salt Lake City temperature, as modeled above.  Of course, omega \+
will still be " }{TEXT 273 5 "Pi/12" }{TEXT -1 282 ", but now you have
 different daily average temperature, amplitude, and phase than was us
ed in equation (2) on page 57, when summer in Georgia was being modele
d.  Fill in the new values below.  Then work out by hand, using the co
sine addition formula, the values you should take for " }{TEXT 275 10 
"a0, a1, b1" }{TEXT -1 10 ", so that " }{TEXT 274 4 "A(t)" }{TEXT -1 
450 " is given by (1) on page 57, and so that the differential equatio
n (3)  also has the correct parameter values. Include in your writeup:
\n# \n# Average Temperature =\n# Temperature variation amplitude = \n#
 Phase delay (was 4 hours in Georgia example) =\n# \n> omega:=Pi/12.0;
  \n> a0:= \n>    #don't forget the semicolons\n> a1:=\n> b1:=\n> A:=t
->a0 + a1*cos(omega*t) + b1*sin(omega*t);\n# \nWe will assume your hou
se is well-insulated, so that k=0.07:\n> k:=0.07;" }}{PARA 0 "" 0 "" 
{TEXT -1 223 "\nd.  Find the solution to (3) with the particular param
eter choices you made above.  You should get a solution function like \+
 (5) on page 57, but reflecting the Salt Lake City temperatures and th
e new insulation parameter. " }}{PARA 0 "" 0 "" {TEXT -1 188 "\ne.  Id
entify the part of your solution which persists as t approaches infini
ty, i.e. the steady periodic solution.  Your formula should have the s
ame character as equation (6) on page 57." }}{PARA 0 "" 0 "" {TEXT -1 
286 "\nf. Write the steady periodic solution in the form of equation (
7) on page 57, so that you can see the time delay for the inside tempe
rature.  You need to use the cosine addition formula.  The text has a \+
discussion on page 184 which may help you.  It is shown there that if
\n            " }{TEXT 271 3 "   " }{XPPEDIT 18 0 "x(t) := A*cos(omega
*t)+B*sin(omega*t);" "6#>-%\"xG6#%\"tG,&*&%\"AG\"\"\"-%$cosG6#*&%&omeg
aGF+F'F+F+F+*&%\"BGF+-%$sinG6#*&F0F+F'F+F+F+" }}{PARA 0 "" 0 "" {TEXT 
-1 24 "then also\n              " }{TEXT 278 2 "  " }{TEXT -1 1 " " }
{XPPEDIT 272 0 "x(t) := proc (t) options operator, arrow; C*cos(omega*
t-alpha) end proc;" "6#>-%\"xG6#%\"tGf*6#F'7\"6$%)operatorG%&arrowG6\"
*&%\"CG\"\"\"-%$cosG6#,&*&%&omegaGF1F'F1F1%&alphaG!\"\"F1F.F.F." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "where th
e right triangle of Figure 3.4.4 page 185 summarizes the relationships
 between A,B, C, and alpha." }}{PARA 0 "" 0 "" {TEXT -1 232 "\ng.  Ass
ume that your heater shut off at midnight, with the inside temperature
 equal to 70 degrees.  Create a plot like Figure 1.5.11 which displays
 the inside temperature and the outside ambient temperature for the ne
xt three days." }}{PARA 0 "" 0 "" {TEXT -1 341 "\nh.  Create a picture
 like Figure 1.5.10, which also includes the slope field for this diff
erential equation, with our Salt Lake City parameters.  Choose initial
 temperatures between 50 and 80 degrees, in  5 degree increments.  Lab
el the maximum and minimum temperature times by hand, on the display p
rintout, before you hand in the project." }}{PARA 0 "" 0 "" {TEXT -1 
189 "\ni.  So, based on your work in this project, how likely do you t
hink it is that the pipes will freeze if the heater is turned off for \+
3 days starting at midnight?  Explain your reasoning.\n\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 3 0" 78 }{VIEWOPTS 1 
1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }