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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 37 "Module 37: Laplace Transform in
ODE's" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
293 " Most students coming into these notes will have had an under
graduate exposure to ordinary differential equations and to the use of
Laplace Transforms for solving these. In this first module on using L
aplace transforms, we recall some of the elementary ideas associated w
ith this subject. " }}{PARA 0 "" 0 "" {TEXT -1 71 " We will exploit
our advantage in having Maple to make computations." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 " Suppose that f(t)
is sectionally continuous on every interval of the form [0, a]. The L
aplace transform of f is defined as" }}{PARA 0 "" 0 "" {TEXT -1 22 " \+
L(f)(s) = " }{XPPEDIT 18 0 "int(exp(-s*t)*f(t),t = 0 .. infi
nity);" "6#-%$intG6$*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF-!\"\"F--%\"fG6#F.
F-/F.;\"\"!%)infinityG" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "We give two simple examples: the L
aplace transform of 1 and of exp(-3 t)." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 92 "assume(s>0);\nint(exp(-s*t)*1,t=0..infinity);\nint(ex
p(-s*t)*exp(-3*t),t=0..infinity);\ns:='s';" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 348 "Of course, we do \+
not have to evaluate integrals for every Laplace transform. Maple know
s many Laplace transforms. We read in the transform package and obtain
access to Laplace transforms. The calling definition is to read in th
e integral transform package. Then, to get the Laplace transform of f(
t) going from the t variable to an s variable, type" }}{PARA 0 "" 0 "
" {TEXT -1 30 " laplace(f(t), t, s)." }}{PARA 0 "" 0 ""
{TEXT -1 18 "Here are examples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 15 "with(inttrans):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
laplace(t,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "laplace(
t^2,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "laplace(sin(a*
t),t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "laplace(cos(a*t
),t,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "Here is a guarantee
that the Laplace transform can be computed: Suppose that f(t) is sec
tionally continuous for every interval of the form [0, a]. If, for so
me constant c, we have " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "limit(exp(-c*t)*f(t),t =
infinity);" "6#-%&limitG6$*&-%$expG6#,$*&%\"cG\"\"\"%\"tGF-!\"\"F--%
\"fG6#F.F-/F.%)infinityG" }{TEXT -1 6 " = 0," }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "then the Laplace transfor
m of f exists for s > c. Such a function is said to be of " }{TEXT
256 17 "exponential order" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1
226 " What are some examples of functions with exponential order? \+
Any bounded, sectionally continuous function has exponential order. Ev
en functions of the form exp( a t) are. A function that is not of expo
nential order is exp(" }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"#" }{TEXT
-1 3 " )," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
19 " The so called " }{TEXT 267 16 "shifting theorem" }{TEXT -1
34 " is suggested in the calculations." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 50 "laplace(exp(b*t)*sin(t),t,s);\nlaplace(sin(t),t,s);"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "laplace(exp(b*t)*cosh(t),
t,s);\nlaplace(cosh(t),t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257
17 "Shifting Theorem:" }{TEXT -1 4 " If " }{XPPEDIT 18 0 "L(F(t),t,s) \+
= f(s);" "6#/-%\"LG6%-%\"FG6#%\"tGF*%\"sG-%\"fG6#F+" }{TEXT -1 7 ", th
en " }{XPPEDIT 18 0 "L(exp(a*t)*F(t),t,s)=f(s-a)" "6#/-%\"LG6%*&-%$exp
G6#*&%\"aG\"\"\"%\"tGF-F--%\"FG6#F.F-F.%\"sG-%\"fG6#,&F2F-F,!\"\"" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0
"" {TEXT -1 5 "Proof" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 58 "This result is easy to understand. The hypothesis is th
at " }{XPPEDIT 18 0 "L(F(t),t,s)=f(s)" "6#/-%\"LG6%-%\"FG6#%\"tGF*%\"s
G-%\"fG6#F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}
{PARA 0 "" 0 "" {TEXT -1 14 " f(s-a) = " }{XPPEDIT 18 0 "int(exp(-
(s-a)*t)*F(t),t=0..infinity)=int(exp(-s*t)*exp(a*t)*F(t),t=0..infinity
)" "6#/-%$intG6$*&-%$expG6#,$*&,&%\"sG\"\"\"%\"aG!\"\"F/%\"tGF/F1F/-%
\"FG6#F2F//F2;\"\"!%)infinityG-F%6$*(-F)6#,$*&F.F/F2F/F1F/-F)6#*&F0F/F
2F/F/-F46#F2F//F2;F8F9" }}{PARA 0 "" 0 "" {TEXT -1 17 "and this last i
s " }{XPPEDIT 18 0 "L(exp(a*t)*F(t),t,s)" "6#-%\"LG6%*&-%$expG6#*&%\"a
G\"\"\"%\"tGF,F,-%\"FG6#F-F,F-%\"sG" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "In using Laplace trans
forms to solve differential equations, it is important to be able to g
o backwards --- to go backwards in the sense that we can find f so tha
t, for example, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 26 " L(f(t), t, s) = " }{XPPEDIT 18 0 "(s+4)/(s^2+3*
s+2);" "6#*&,&%\"sG\"\"\"\"\"%F&F&,(*$F%\"\"#F&*&\"\"$F&F%F&F&F*F&!\"
\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 32 "Of course, Maple does this, too." }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "invlaplace((
s+4)/(s^2+3*s+2),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 ""
}}}{PARA 0 "" 0 "" {TEXT -1 150 "You may recall that humans typically \+
found the inverse Laplace transform of the previous example by first p
erforming a partial fraction decomposition." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "convert((s+4)/(s^2+3*s+2),parfrac,s);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 59 "Th
en, a table look up for the inverse Laplace transform of " }{XPPEDIT
18 0 "1/(s+2);" "6#*&\"\"\"F$,&%\"sGF$\"\"#F$!\"\"" }{TEXT -1 9 " and
of " }{XPPEDIT 18 0 "1/(s+1);" "6#*&\"\"\"F$,&%\"sGF$F$F$!\"\"" }
{TEXT -1 106 " , or remembering these from past experiences would achi
eve the Laplace transform of the original problem." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 38 "map(invlaplace,[1/(s+2),1/(s+1)],s,t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 172 " From the perspective of differential equations, the most \+
important notion for Laplace transforms is what happens when we take t
he Laplace transform of y ' and of y ''." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 256 "" 0 "" {TEXT 258 323 " Because we will want to s
olve differential equations by the methods of Laplace transforms, you \+
can imagine that we will take the Laplace transform of both sides of a
differential equation and perform some algebraic operations. Thus, we
will need to know what is the Laplace transform of the derivative of \+
a function. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "laplace(dif
f(f(t),t),t,s);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 ""
{TEXT 259 56 "How do humans think of this? Use integration by parts: \+
" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "int(exp(-s*t)*diff
(f(t),t),t=0..infinity)" "6#-%$intG6$*&-%$expG6#,$*&%\"sG\"\"\"%\"tGF-
!\"\"F--%%diffG6$-%\"fG6#F.F.F-/F.;\"\"!%)infinityG" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "exp(-s*infinity)*f(infinity)-1*f(0)+s*int(exp(-s*t)*
f(t),t = 0 .. infinity);" "6#,(*&-%$expG6#,$*&%\"sG\"\"\"%)infinityGF+
!\"\"F+-%\"fG6#F,F+F+*&F+F+-F/6#\"\"!F+F-*&F*F+-%$intG6$*&-F&6#,$*&F*F
+%\"tGF+F-F+-F/6#F>F+/F>;F4F,F+F+" }{TEXT -1 1 "." }}{PARA 256 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "If f has exponential o
rder, " }{XPPEDIT 18 0 "exp(-s*infinity)" "6#-%$expG6#,$*&%\"sG\"\"\"%
)infinityGF)!\"\"" }{TEXT -1 3 " f(" }{XPPEDIT 18 0 "infinity;" "6#%)i
nfinityG" }{TEXT -1 38 ") = 0. This the same result Maple got." }}
{PARA 256 "" 0 "" {TEXT 260 2 "\n\011" }{TEXT 271 137 "Naturally, we a
re not interested in solving only first order equations. What happens \+
with Laplace transforms of higher order derivatives?" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 39 "laplace(diff(F(t),t,t),t,s);\nexpand(%);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 45 "Laplace transforms and differential equations" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}}{PARA 0
"" 0 "" {TEXT -1 31 " We examine a few problems." }}{PARA 0 "" 0 "
" {TEXT 264 10 "Problem 1:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }{TEXT 265 91 "We solve this differential equation two ways: \+
using dsolve, and using laplace transforms: " }{TEXT 270 25 " \+
" }}{PARA 0 "" 0 "" {TEXT 272 69 " \+
y'' + 3 y' + 2y = 0, y(0) = 1, y'(0) = 0.\n" }{TEXT 269 16 "(1)
Using dsolve" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "dsolve(\{di
ff(y(t),t,t)+3*diff(y(t),t)+2*y(t)=0,y(0)=1,D(y)(0)=0\},\n\011\011y(t)
);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 266 85
"(2) Using laplace, first recall what is the Laplace transform of y'(t
) and of y''(t)." }{TEXT 268 1 " " }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 64 "laplace(diff(y(t),t),t,s); expand(laplace(diff(y(t),
t,t),t,s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h(s)=solve(s
^2*h(s)-s*1-0 + 3*(s*h(s)-1) + 2*h(s)=0,h(s));" }}}{PARA 0 "" 0 ""
{TEXT -1 86 " Thus, the problem is to find the inverse Laplace tra
nsform of this last function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 23 "invlaplace(rhs(%),s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 56 " This should be the same an
swer that Maple got with " }{TEXT 273 6 "dsolve" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 10 "Problem \+
2:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 73 " We solve the differential equation with a non-zero
forcing function." }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT
18 0 "diff(y(t),t)+3*y(t)=1" "6#/,&-%%diffG6$-%\"yG6#%\"tGF+\"\"\"*&\"
\"$F,-F)6#F+F,F,F," }{TEXT -1 16 ", with y(0) = 0." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "dsolve(\{diff(y(t),t)+3*y(t)=1,y(0)=0\},y(t)
);" }}}{PARA 0 "" 0 "" {TEXT -1 61 "Here is how we would do this probl
em with Laplace transforms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
27 "ode:=diff(y(t),t)+3*y(t)=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 17 "laplace(ode,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
15 "subs(y(0)=0,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solv
e(%,laplace(y(t),t,s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "
invlaplace(%,s,t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 274 10 "Problem 3:" }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 " We solve the same differenti
al equation, but this time, we have a non constant forcing function,"
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "ode:=diff(y(t),t)+3*y(t)=H
eaviside(t-1)-Heaviside(t-2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 17 "laplace(ode,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "
subs(y(0)=0,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(%,
laplace(y(t),t,s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simp
lify(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invlaplace(%,s,
t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ysol:=unapply(%,t);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(\{Heaviside(t-1)-H
eaviside(t-2),ysol(t)\},t=0..7);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT
261 1 "\n" }{TEXT 262 10 "Problem 4:" }}{PARA 0 "" 0 "" {TEXT -1 84 " \+
We solve a second order differential equation with a periodic forc
ing function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y:='y';" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ode:=diff(y(t),t,t)+3*diff(y(t),t)+
2*y(t)=sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "laplace(o
de,t,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{y(0)=0,D
(y)(0)=0\},%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(%,l
aplace(y(t),t,s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "invla
place(%,s,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ysol:=unap
ply(%,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(\{sin(t),
ysol(t)\},t=0..4*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 11 "Assignment:" }{TEXT -1
63 " Overlay the graphs for the solutions of these three problems:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(1) y '
' + 4 y ' + y = 0, y(0) = 1, y '(0) = 2," }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(2) y '' + 2 y ' + y = 0, y(0) = \+
1, y '(0) = 2," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 48 "(3) y '' + 1 y ' + y = 0, y(0) = 1, y '(0) = 2," }}}
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