CHAP.
ll THE LAPLI\CE TRANSFORM
SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS
In the following list of theorems v/e assume, unless otherwise stated, that all functions
satisfy the conditions of Theorem 7-7 so that their Laplace transforms exist.
1. Linearity ProPertY.
Theorem 7-2. If cr and cz&tE any constants while Fr(t) and Fz(t) arc functions
with Laplace transforms /r(s) and lz(s) respectively, then
4{erFit') + czFz(t\) : cr4{Iv'r(t)} + cz({F'r(t)} = cr^(s) + c2f2(s) (2)
Theresultiseasilyextendedtornorethantwofunctions.
Bxample. 4utz - 3cos2f * = 44-{t2) - 3{{cos?t} * 6"({e-t'l.
'e-t\
- 41-2j\ * \ -r b/-1 \
-*/"\E'z+4/
\*/ "\s*1/
trons-
The symbol .(, which transforms F(t) into /(s), is often caJled the Laplnce
f ormation operatir. Because of the
property of .( expressed in this theorem, we say
that *( is a knear operator or that it has the lineari,tE property.
2. First translation or shifting property'
Theorem 7'3. If "( {f'(4} = l(s) then
-( {e"t F(t'1} : f(s - o) (3)
Example. Since { {cosZt} : -r+7, we have
4{e-tcoszt) = Cf#t7 = ;z++=
3. Second translation or shifting property.
Theorem l-tt. If { {F(t)} = l(s) and G(t) : {ot'-t 'r'.i, then
{ {G(4} : €-^l(s) (4)
Example. Since {{ts} = # = 3, the Laplaee transform of the function
G(t) = {f - r' iai
i= 6"_zstst.
4. Change of scale ProPertY-
Theorem 1-5. If { {F'(4} = l(s), then
^({r'(ot)} = }t(i) (5)
ExamPle. Since {{sint} = *h, we have
-( {sin 3t} = + it# .l = *+T '
THE LINEARITY PROPERTY
5. Prove the l;i'nearitg propertE lTheorem 1-2, Page 3l'
I€t {{fr(t)} = /r(e) -- Ir- e-"t4tflldt and {{F,r1t)1 = /z(s) = I- e-3t4zftld't' Then if
c1 and c2 are anY constents,
{c1tr,1(f) l cz4zfll}
= c, e-"t F{t) d,t + a,
f* e-'t F2Q) itt
f
o14{F1(t)} I azl{Fz0l}
= otf:(al + czf|kl
The result is 6asily generalized [aee Problem 61]'
CHAF.ll THE LAPLACE TRANSFORM t3
6. Find 4.{4et' + 6f - 3 sin4t }- ?cosZt\.
By the linearity property [Problem 5] we have
.t{4e3t + 6t3 - 3sin4t *2cos2t} : 4-C{e5t} + 6"({ts} - 8{{sin4t} + 2.{{cos2r}
,(*) .'(*u) -'(o+ie) *,(t'-)
436LZ2s
;*b -r s+ - ;4 16
T 72+4
where s ) 5.
TRANSLATION AI{D CHANGE OF SCALE PROPERTIES
7. Prove the fi.rst translation, m sldfting propertE: If ^( {F'(r)} : l(s), then "( {et F(t)\ :
l(s - o).
'We have { {F,(D} = f o' "-* rld a' : /(s)
Then { {e"t F(t)} = f o* "-u l*t F(t)l d't
/
= f n' "- r"-or F(t't ilt = /(s - a)
8. Find (a) -q ltz est\, (b) { {e-2t sin 4t\, (c'1 -( {aat eosh 5t\ , (d\ { {e-zt (3 cos 6t - 5 sin 6t)}.
(o) <{tz) = # = 3. rhen 4{tzsetY: CirF.
' (b) {{sin4r} : *+u. Then 4{e-ztsin4f} = GTr=r+16 = p+**,O'
(c) -({cosh rtl : V\6. Then -ql4tcoshs* : -;-S;+--='
ffi
Anather method.
12<
br} = . =
{", (y#)}
.c {eat cosh {a' + "-')
tl1
= t1s-e-'+1frI t-4
;t-&-e
(d) {{3cos6t - 5sin6f} = 3{{cos6i} - 6"({sin6t}
^/ e \ -/ o \- 3s-30
- "\sz*36/ "\s2+36/ s2+36
Then l{e-zt(Bcos6r-6sinct)} = ffi = ffi
t4 TIIE LAPI^ACE TRANSFORM lclraP. 1
9. Prove the seeand, translntian or shifting prapertg:
If a{F(r)} =/(s) and G(r) : {ftt-' ia:, then {{c(D} : e-""f(s).
4.{G(t',) = fo* "=*ep1at
: fo" "-uci(t}
dt + ["- e-"tG(t) d.t
= to" u-nro, o, + f- "-ur$-a\ d,t
= t e-"t F(t - a1 dt
"*
lr*
: e-stu+o F(u) du
)o
= ,-* J6
f* e-ilF(u\ itu
f(s)
"-os
where we have used the substitution f = ,*:.
r0. Find { {r(r)} ir F(t) : {;"',t - 2'/3) iafl',?
Methott 1. { {r(r)} = f o'"'" "-u ,0, o, + f :,".-"rcos (r - 2r/B) d,t
= fo* "-"ru**/$
cosu d,u
= e-ztsla
fo* "-",
eosu du = #
Metlwd 2. Sinee { {cos t} = F}1, it follows from Problem g, with a : 2c/8, that
{ {r'(t)} = "":"'"=''
11. Prove the change of scale property: If {{f'(t)} = l(s), then ^({r'(ot)} : }t(i)
.({F(at)} = fo' "-*Fqntl at
= f o "-"ru,o, F(u) it(ula)
= !fr* e-*r"r@) du
= fr(s)'r' \a /
ueing the transfomati on t = u/a,
CHAP,ll rI{E,LAPLACE TRANSFORM 15
12. Given that *t"Oi, tan-r(l/s), find -{-t}
):
By Problem 11,
: :*{*, '} = },",,-' {t/(s/at} = L611-t(a/s)
{=f}
rhen -{=t} - .*,'-r (a/a).
