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{SECT 0 {PARA 263 "" 0 "" {TEXT -1 52 "Tutorial: Solving Differential \+
Equations Using Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 14 "James V. Herod" }}
{PARA 257 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 258 ""
0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 259 "" 0 "" {TEXT -1 22 "Atlan
ta, Georgia 30332" }}{PARA 260 "" 0 "" {TEXT -1 3 "USA" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 8 "Abstract" }}{PARA 0 "
" 0 "" {TEXT -1 340 "Users of Maple find that the program provides a v
ariety of methods to obtain solutions for differential equations. Ther
e are both analytic and numerical methods. Further, the complexity of \+
equations that can be solved has increased with each new release of Ma
ple. This tutorial will give an introduction to some of the techniques
available." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "
" {TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 281 " A study of \+
differential equations typically begins in the calculus. For science a
nd engineering students, differential equations continue to be importa
nt throughout the undergraduate education. These students use differen
tial equations as a standard tool for creating models. " }{TEXT -1
222 "Differential equations remain important in the research and devel
opment performed by mature scientists and engineers. In this tutorial,
we present techniques for using Maple to compute solutions for differ
ential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 44 "General solutions for differential equations" }}
{PARA 0 "" 0 "" {TEXT -1 170 " An introductory discussion of diff
erential equations usually begins with a simple equation for which one
finds a general solution. A nonhomogeneous equation such as" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " \+
" }{XPPEDIT 18 0 "diff(y(t),t)+y(t)=sin(t)" "6#/,&-%%diffG6$-%
\"yG6#%\"tGF+\"\"\"-F)6#F+F,-%$sinG6#F+" }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 145 "is a simple equation for which a ge
neral solution can be found. Using Maple, we input the equation as fol
lows, and simply ask for the solution y(" }{TEXT 264 1 "t" }{TEXT -1
14 ") with dsolve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "de:=di
ff(y(t),t)+y(t)=sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
sol:=dsolve(de,y(t));" }}}{PARA 0 "" 0 "" {TEXT -1 94 " The genera
l solution for this equation is the right-hand-side of what we have ca
lled sol." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Ygen:=simplify(
rhs(sol));" }}}{PARA 0 "" 0 "" {TEXT -1 51 " It is not hard to verify \+
that Ygen is a solution: " }{XPPEDIT 18 0 "e^(-t)" "6#)%\"eG,$%\"tG!\"
\"" }{TEXT -1 42 " is a solution of the homogeneous equation" }}{PARA
0 "" 0 "" {TEXT -1 17 " " }{XPPEDIT 18 0 "diff(y(t),t)
+y(t) = 0" "6#/,&-%%diffG6$-%\"yG6#%\"tGF+\"\"\"-F)6#F+F,\"\"!" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 52 "and one solution for th
e non-homogeneous equation is" }}{PARA 0 "" 0 "" {TEXT -1 18 " \+
" }{XPPEDIT 18 0 "(sin(t)-cos(t))/2" "6#*&,&-%$sinG6#%\"tG\"
\"\"-%$cosG6#F(!\"\"F)\"\"#F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 " We can also ask Maple t
o verify that Ygen is a solution by taking the derivative and doing th
e arithmetic." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(di
ff(Ygen,t)+Ygen = sin(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 41 "Initial Value and Boundary Value Problems" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 458 " More often, on
e will wish to solve a differential equation with initial conditions o
r boundary conditions. Plotting the solutions for a variety of initial
conditions gives a sense of the flow associated with the equation. Fo
r this illustration, we take the same equation as above, but plot thre
e solutions starting at y(0) = 0, y(0) = 1, and y(0) = -1, respectivel
y. Superimposing the graphs of the solutions onto the graph of the for
cing function, sin(" }{TEXT 261 1 "t" }{TEXT -1 74 "), gives a visuali
zation for the effect of this periodic right-hand-side. " }}{PARA 0 "
" 0 "" {TEXT -1 58 " We solve the equation once with a symbolic pa
rameter " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 20 " and substitute f
or " }{TEXT 257 1 "a" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol:=dsolve(\{de,y(0)=a\},
y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "y1:=subs(a=0,rhs(
sol)):\ny2:=subs(a=1,rhs(sol)):\ny3:=subs(a=-1,rhs(sol)):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(\{sin(t),y1,y2,y3\},t=0..2*Pi)
;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 501 " \+
We repeat this illustration of how solutions for a differential equ
ation change, but this time we choose a variety of boundary conditions
. Being able to solve a second order differential equation with bounda
ry conditions, instead of only initial conditions, is a standard featu
re of Maple. In this example, the forcing function will be an impulse
over an interval as defined by a difference of step functions. We def
ine the step functions by using the Heaviside function. The equation i
s " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "g:=t->5*(Heavis
ide(t-1/3)-Heaviside(t-2/3)):\nde:=diff(y(t),t,t)+3*diff(y(t),t)+2*y(t
)=g(t);" }}}{PARA 0 "" 0 "" {TEXT -1 106 " We solve for three boundary
conditions, simplify the first solution, and plot the three solution \+
curves. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "y1:=dsolve(\{de
,y(0)=0,D(y)(0)=0\},y(t)):\ny2:=dsolve(\{de,y(0)=0,y(1)=0\},y(t)):\ny3
:=dsolve(\{de,y(0)=0,y(1)=1\},y(t)):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 31 "collect(y1,Heaviside,simplify);" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 63 "plot([rhs(y1),rhs(y2),rhs(y3)],t=0..1,color=[
red,black,green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "F
orcing Functions from Data." }}{PARA 0 "" 0 "" {TEXT -1 299 " In app
lications, one may be confronted with data from which to define the fo
rcing function. From the data, models are constructed and examined usi
ng the tools of analysis. We illustrate how one might start with a col
lection of data and construct a forcing function for the differential \+
equation. " }}{PARA 0 "" 0 "" {TEXT -1 79 " Suppose the forcing fu
nction is determined from the following data points:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 " \+
h(0) = 1, h(1) = 1, h(2) = 4, h(3) = 3, and h(4) = 1. " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "We ask for a solutio
n to the equation " }{TEXT 262 20 "y'(t) + y(t) = h(t) " }{TEXT -1 5 "
with " }{TEXT 263 8 "y(0) = 0" }{TEXT -1 102 ". The first task will be
to fit the data with an analytic function. To accomplish this task, w
e use a " }{TEXT 256 10 "spline fit" }{TEXT -1 61 " to construct a for
cing function corresponding to this data. " }}{PARA 0 "" 0 "" {TEXT
-1 6 " A " }{TEXT 258 12 "cubic spline" }{TEXT -1 456 " defines a f
unction that has a continuous derivative and that goes through each da
ta point. One needs four pieces of information to define the four cons
tants for a cubic. Requiring that the cubic should go through two data
points will not be enough information to determine four coefficients \+
for the cubic polynomial. Cubic splines are required to have first and
second derivatives that agree with the cubic polynomials connecting t
he adjacent points. The " }{TEXT 259 19 "natural spline fits" }{TEXT
-1 106 " are commonly defined by requiring that the cubics at the extr
eme ends should have second derivative zero." }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "xcoo
rd := [0,1,2,3,4]:\nycoord := [1,1,4,3,1]:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "h:=spline(xcoord,ycoord,x,cubic);" }}}{EXCHG {PARA
0 "" 0 "" {TEXT -1 184 "To see the spline fit, we plot the spline and \+
the data together on the same graph. To do this we create two separat
e plots then display them together using the plots[display] command."
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "J:=plot(h,x=0..4): \nK:=
plot([seq([xcoord[i],ycoord[i]],i=1..5)],\nstyle=POINT,color=BLACK):\n
plots[display](\{J,K\});" }}}{PARA 0 "" 0 "" {TEXT -1 89 "This spline \+
fit for the data is used as the forcing function for a differential eq
uation." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sol:=dsolve(\{dif
f(y(x),x)+y(x)=h,y(0)=0\},y(x));" }}}{PARA 0 "" 0 "" {TEXT -1 135 "To \+
see the result graphically, we graph the solution for the differential
equation superimposed with the graph of the forcing function." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "L:=plot(rhs(sol),x=0..4,colo
r=BLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display](
\{J,K,L\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 44 "Directi
on Fields for a Differential Equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 307 " With tools such as Maple to red
uce the difficulty in graphing solutions for differential equations, t
he emphasis could turn to the qualitative behavior of equations. For t
his purpose one can draw direction fields in order to see the flow of \+
solutions. It is instructive to draw some typical solutions. " }}
{PARA 0 "" 0 "" {TEXT -1 96 " To illustrate these techniques, we \+
draw the direction field for a simple logistic equation" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " " }
{XPPEDIT 18 0 "diff(y(t),t) = y(t)*(1-y(t))" "6#/-%%diffG6$-%\"yG6#%\"
tGF**&-F(6#F*\"\"\",&F.F.-F(6#F*!\"\"F." }{TEXT -1 1 " " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "and superimpose the
solution for this equation beginning at different initial values. Fir
st, we simply draw the direction field." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "de:=diff(y(t),t)=y(t)*(1-y(t
));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dfieldplot(de,y(t),t
=0..2,y=-1..2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 88 " We construct solutions starting at several initial v
alues and plot these solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 30 "sol:=dsolve(\{de,y(0)=a\},y(t));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 79 "y1:=subs(a=1/3,rhs(sol));\ny2:=subs(a=5/3,rhs(sol)):
\ny3:=subs(a=-1/10,rhs(sol)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 32 "plot(\{y1,y2,y3\},t=0..2,y=-1..2);" }}}{PARA 0 "" 0 "" {TEXT -1
78 "\nFinally, we superimpose the graphs of the solutions with the dir
ection field." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "J:=dfieldp
lot(de,y(t),t=0..2,y=-1..2):\nK:=plot(\{y1,y2,y3\},t=0..2,y=-1..2,colo
r=BLACK):\nplots[display](\{J,K\});" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{SECT 1
{PARA 3 "" 0 "" {TEXT -1 55 "Direction Fields for a System of Differen
tial Equations" }}{PARA 0 "" 0 "" {TEXT -1 452 " Systems of equati
ons arise in all disciplines of science and engineering. For example, \+
population models, compartment models, and chemical reaction equations
are examples of applications for systems of equations. In addition to
being able to solve many of these analytically, direction fields can \+
be drawn for two dimensional systems. Just as with one dimensional sys
tems, these direction fields for systems give a visualization for the \+
dynamics. " }}{PARA 0 "" 0 "" {TEXT -1 55 "We give the direction field
s for the following system: " }{TEXT 265 1 "x" }{TEXT -1 4 "' = " }
{XPPEDIT 18 0 "1-y^2" "6#,&\"\"\"F$*$%\"yG\"\"#!\"\"" }{TEXT -1 3 ", \+
" }{TEXT 266 1 "y" }{TEXT -1 4 "' = " }{XPPEDIT 18 0 "x+2*y" "6#,&%\"x
G\"\"\"*&\"\"#F%%\"yGF%F%" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 51 "deq1:=diff(x(t),t)=1-y^2;\ndeq2:=diff(y(t),t)=x+2*y;
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 " \+
Curves will be drawn corresponding to a variety of initial values for
x and y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 125 "inits:=\{[0,-3,3],[0,-1,3],[0,1,3],[0,3,3],\n\011
\011\011[0,-3,-3],[0,-1,-3],[0,1,-3],[0,3,-3],\n\011\011\011[0,-3,1],[
0,-3,-1],[0,3,1],[0,3,-1]\};" }}}{PARA 0 "" 0 "" {TEXT -1 102 "As a re
minder that the DEplot command is found in the DEtools, package, we re
ad in this package first." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
102 "with(DEtools):\nDEplot([deq1,deq2],[x,y],t=-5..5,inits,\n\011step
size=0.1,x=-4..4,y=-4..4,linecolor=BLACK);" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 286 " It can be seen from the f
ield plot that there are likely two points of equilibrium. We find the
se analytically by solving for where the derivatives of x and of y are
zero. Almost, we could guess the result of the next calculation by lo
oking at the above graphs and direction field." }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 31 "solve(\{1-y^2=0,x+2*y=0\},\{x,y\});" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 ""
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Numerical Methods." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 264 " It is rather \+
typical in science and engineering that solutions for differential equ
ations cannot be found in analytic form. In this case, one might use n
umerical methods for finding solutions. We illustrate how this is done
in Maple with a classical equation. " }}{PARA 0 "" 0 "" {TEXT -1 85 "
Students study models for a frictionless pendulum. This is most o
ften modeled as" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "de := dif
f(theta(t),t,t)+g/l*sin(theta(t)) = 0: de;" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "To simplify the model, it is of
ten assumed that " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 25 " \+
is small. In that case, " }{XPPEDIT 18 0 "sin(theta)" "6#-%$sinG6#%&th
etaG" }{TEXT -1 27 " is approximately equal to " }{XPPEDIT 18 0 "theta
" "6#%&thetaG" }{TEXT -1 31 " so the equation is changed to" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "des := diff(theta(t),t,t) + \+
g/l*theta(t) = 0: des;" }}}{PARA 0 "" 0 "" {TEXT -1 263 "How solutions
for the nonlinear equation compare with linearized model can be illus
trated by computing solutions for the nonlinear equation numerically \+
and superimposing the graph of this solution with the solution of the \+
simpler equation. We arbitrarily choose " }{XPPEDIT 18 0 "g/l" "6#*&%
\"gG\"\"\"%\"lG!\"\"" }{TEXT -1 55 " = 1 and use Maple to solve both e
quations numerically." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "T := dsolve(\{subs(g=l,de),\ntheta
(0)=Pi/4,D(theta)(0)=0\},theta(t),\ntype=numeric,output=listprocedure)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The output is a list of proc
edures which evaluate numerically. We first select the procedure for \+
" }{XPPEDIT 18 0 "theta(t)" "6#-%&thetaG6#%\"tG" }{TEXT -1 22 " then \+
evaluate it at " }{XPPEDIT 18 0 "t=0.1 and t=0.2" "6#3/%\"tG$\"\"\"!\"
\"/F%$\"\"#F(" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 20 "T:=subs(T,theta(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Th
e numerical values for y(t) are now computed using a Runge-Kutta 45 al
gorithm -- more sophisticated algorithms are available. " }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "T(0.1), T(0.2);" }}}{EXCHG {PARA 0
"" 0 "" {TEXT -1 64 "Next we solve the simplified DE similarly and plo
t both together" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "TA:=dso
lve(\{subs(g=l,des),\n theta(0)=Pi/4,D(theta)(0)=0\},\n theta(t),\n \+
type=numeric,output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 24 "TA := subs(TA,theta(t));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 66 "plot(\{T,TA\},0..6*Pi,title=\n`Pendulum solution an
d approximation`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 146 " The extent to which these separate can \+
now be explored by increasing the initial value, or extending the inte
rval over which the graphs are made." }}}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 45 "Analytic Solutions Using Integral Transforms." }}{PARA 0
"" 0 "" {TEXT -1 322 " Integral transforms provide a common tool f
or solving differential equations. The methods are particularly approp
riate for solving differential equations with discontinuous, or period
ic forcing functions. The Laplace transform of a function F(t) is defi
ned to be a function f(s) with the following integral transform:" }}
{PARA 0 "" 0 "" {TEXT -1 41 " f(s) = L(F)(s) = \+
" }{XPPEDIT 18 0 "int(exp(-s*t)*F(t),t=0..infinity)" "6#-%$intG6$*&-%$
expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F--%\"FG6#F.F-/F.;\"\"!%)infinityG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 431 " These transforms change the job of solving many par
tial differential equations to one of solving an ordinary differential
equation. In turn, Laplace transforms change ordinary differential eq
uations to algebraic equations. For this reason, one might observe fro
m time to time that Maple will solve a large system of linear, ordinar
y differential equations faster by the techniques of Laplace transform
s than by not using them." }}{PARA 0 "" 0 "" {TEXT -1 127 " We ill
ustrate the techniques for asking Maple to use Laplace transforms by g
iving a differential equation for which Maple " }{TEXT 260 5 "needs" }
{TEXT -1 40 " the Laplace techniques. The equation is" }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " \+
" }{XPPEDIT 18 0 "diff(y(t),t) + y(t) = 1- int(y(s),s=0..t)" "6#/,&-%%
diffG6$-%\"yG6#%\"tGF+\"\"\"-F)6#F+F,,&F,F,-%$intG6$-F)6#%\"sG/F5;\"\"
!F+!\"\"" }{TEXT -1 16 ", with y(0) = 0." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "We solve the equation with the met
hod of Laplace transforms and plot the solution." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "deq:=diff(y(t),t)+y(t)+int(y(s),s=0..t)=1;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "diffint:=dsolve(\{deq,y(0)=
0\},y(t),method=laplace);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
18 "Ydi:=rhs(diffint):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p
lot(Ydi,t=0..10);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 174 " Finally, in [1 ], Monagan and Lopez discussed the g
eneration of periodic functions and point out that one might want to t
ake the Laplace transform of a function such as " }{XPPEDIT 18 0 "t-fl
oor(t)" "6#,&%\"tG\"\"\"-%&floorG6#F$!\"\"" }{TEXT -1 158 ". Recalling
(See [2].) that if F is a bounded, piecewise continuous function defi
ned for positive numbers and has period T, then the Laplace transform \+
of F is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
35 " " }{XPPEDIT 18 0 "int(exp(-s*t)
*F(t),t=0..T)/(1-exp(-s*T))" "6#*&-%$intG6$*&-%$expG6#,$*&%\"sG\"\"\"%
\"tGF.!\"\"F.-%\"FG6#F/F./F/;\"\"!%\"TGF.,&F.F.-F)6#,$*&F-F.F7F.F0F0F0
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 40 "We can compute the Laplace transform of " }{XPPEDIT 18 0
"t-floor(t)" "6#,&%\"tG\"\"\"-%&floorG6#F$!\"\"" }{TEXT -1 12 " as fol
lows:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Transform:=int(exp(
-s*t)*t,t=0..1)/(1-exp(-s*1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 20 "simplify(Transform);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 101 "Having the Laplace transform o
f this periodic function, solutions for differential equations such as
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " \+
" }{XPPEDIT 18 0 "diff(y(t),t) + y(t)) = t - floor(
t)" "6#/,&-%%diffG6$-%\"yG6#%\"tGF+\"\"\"-F)6#F+F,,&F+F,-%&floorG6#F+!
\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "a
re accessible." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 11 "References:" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "[1] Michael B. Mo
nagan and Robert J. Lopez, Tips for Maple Users and Programmers, " }
{TEXT 267 9 "MapleTech" }{TEXT -1 36 ", Vol.3, No. 3, pp. 10 - 17, (19
96)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "[
2] Ruel V. Churchill, " }{TEXT 268 23 "Operational Mathematics" }
{TEXT -1 50 ", Third Edition, McGraw-Hill Book Company, (1972)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
262 "" 0 "" {TEXT -1 17 "Authors Biography" }}{PARA 0 "" 0 "" {TEXT
-1 557 "Jim Herod is a professor of mathematics at Georgia Tech in Atl
anta, Georgia. His interests have been in mathematical physics, mathem
atical biology, and the structure of differential equations. Early in \+
his career, he worked with a team in creating a model for geo-magnetic
ally trapped particles and, more recently, with a team studying models
for the Boltzmann Energy Equation. His collaborations on the structur
e of differential equations produced a Hille-Yoshida theory for evolut
ion operator equations. During the past year, he co-authored a book ti
tled " }{TEXT 270 75 "An Introduction to the Mathematics of Biology, w
ith Computer Algebra Models" }{TEXT -1 1 "." }}}{MARK "0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }