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-
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J;
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a
(r*s1:
:) s-*.ns t r","lJ- C- "i,+*;
ptt?k'.-
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-4
c(9 =
4
F(i-
I
q.,
JJ-" f, {6ttt = t's^ft"t
pndfi- 5e reve -
-
1 1GLr)J= [i .t't.au>,at.
= .f3;'t atttat *J" ;'t ato:+
= [i "'tO:at
+
J"
;'t nG,"
o +
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6
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d€ h4* 11t3J . 3l
sr
L
t tJ- 5l
L
Ld ur^d idl- ptuiuZ ! at &ve -
1{ e'Fti} =
Let t- a : u.,." -2t:
-It^'A
!'a A ,6? k^rz-
-1'{ 6Lt8
O) c.nde
r= a]
t=' I!r:a)
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= f" ;4'*D ,,u1su
J.
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= 2 t)Je) .
{ sd€ Pn+."fu'.-
Y{ L{ F(ol=f (") ,
lz--or,l
, ra -(1)"
+J(*).
Ft4{{ =
d*"1{
BC' ou
"^"..-1,
{F(^t)]
=
|"."-st
Frq)Aa
..o
Tr^^nln O. o. t',." -
-rS
7
qr,"d.h"C.,€fs..r"
S c"s"t
p""r".t , .f.r {.4 L^pr.-
J"+1
ft*l;'
-
1,! c.:t] =JG)
s
+!(+)r1.ic..^tl=
-L
3/^
J.,l'4
a lu^r $ IIFOI. (rs), +z,v
zlr"r+tJ. - sF(0
Pv&- '0
,,t p1oJt
1;frad- f'
^ --Le Ftt/arFrdl+sl
to) -f sJ(5,
- F(,) .
ddr 4 L"?ncz
ri;q.{
= sltO
ffc.)
(F) t-.plq.. 1*"$'r d
j :f 'a';. {ro ,tL"-l,in?oJ--sf(o-F1")
Pn',xft-bl L.gcc 1n""*f'n. ae t".t-
1iF(Dj. Jn = drt n(rttt
e
tt
r 'ct):t r
J:
-Jt " tlt
- e'o).
- t(')
x.1r"'(E=
"io-
'' -1"i F"Lt/| ott r"(t):t
'P-'*r'c]ll.'
=a-F'u)+s
5
&"hn"i.6 thc P't'ce)o' 6e c'^
- {-' r(o)- s"'"
c l"o)
o
(d L4pr^cei."J"- d h+'Arn):-
19:3-
"
g Tlsc)l=JG) ti4" 1iJ: rG)5
FLu)1."Lo.+ &(t)
r'"." d (t)
c'(o)
lF(ol-er?6)- r'?')
F'6)
-
Ftt)
r,,o)_.-.-f"-'La)
=J:
J."ttot'a'
S'e'rF'OAt
1| rtu|
l,sc"r- rt"l]
(,) - F'(o) .
= - r'co)n
={!G)-s
]-
J:
,'t*t
leto
a'o]=
- 5"-l
FLqduJ
=
1 I F'd)l
P,6. fl
eia
N'o .1{
a L {ru>}'
, s'Jc)
slu)fu =a
I lt rs;kl
{rs)
5
s1.{ {t rt"r&l
l{l'F(u)dqf'
tvo
i
i./er",a4 t^l^",ti^"rJi,.( M,r,'dr.]i"h
ft o*j9:
- $1,!Ftt]'J(t, #.'j
r if rcu]=err"S"rc"r=r,j,.:,:=
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*J! rt ecD
-r,4'
t
Lerrhl+!,s
#"rto
(G) reri
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d* F[0..dt=1"fo.r
t", Atfeh^h',A,^d.il.rxr t"+ftdr6{^.
Jt
rule
f' n=x
tt4
I
" J"ft ;"'ornat
= f'- t;'trto dt
Jo
= - [' a'titrrb]J" r
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a a r.-,..1- )t
'^(, L 11l(r)[
= -:r
-
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-n 6{^tris!, tL, rL-,.- i.
i,v,a.,
. o€
lnl(ciri .1rE(--l't {lat 1ir '{r.uvea
J,"*i-:
(s) -@
'.-.1
FoJdf
-rr^ +(o ust;r" pldJt =
, -J:t', lt..' F(e]rt
> fru,r t"-' rtO]lt
.J"
€d jG-',1
=tf 1t*u,,
" CnJ-'Jq'tctr
tr
rt
tf((e
gllrost'J S O '.'
+'* '" d1t-
ia +",," r,' +e fleora t"'1lr
r-" A tU ll^ow tt ku lt
1t "
n: {+r =2 @vr 't"
1,.zTi^6..' u +""a +' 'U
p*a -1,t t' "*+J

c4.xl =
.r--_+4-
r-,P-1lE- r-. ts. LJ-i47
"13
_ 2ts
-@* r;s
+tu*.^ l"1/,l$' . =t1
&' n =x+'l
.+L Jl-;' il h
3 *J s'-
p"silv" i^1"4t) vJuto
=!@ (t4)
A^.F(t)1 - r!
=J -.1"'d"
T
(11) l;"ist"n
{Fc)l ={(s)
-F(9t
tG
et
JG)
&tr)
F(r)
11F(
to
- JG)
s fn,- s t' ,a,
- I +(r) dt
J3
-- l +Cr)dt
JlctUF .l {r)/r
I 3r5
{c,(Dil')
E4 L"t
tr)
It *.tol
*1{Grej
oJ=)
)
) 4l icrol
l*.da"i'd t g sita d.r.+a
Ltr',rJ:=
f+
9ia
.li61I{o'to1
u?1oc.1=3
11s*^
0o= o
!$) l!ott +tikl,I =f
'),t-Ltfrt,t.?t,rl'
teJ = | foat
=f'oJ'
=
), t t";da
-)
)z
e'tJ]
r.f rl9 r-
X{ettt]=
f"
+T)
1a*. rJ
Oe
J]
{i+ {T
flr)t = I
=it
-!,
-T)o
F(o2t
$ro !d
1l F
+
r)
6e
FI
t ?T-st
*
)","'lp(oJt+f{
d4!"tk" 4
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=1"
Ftt)
otz. Etl
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i'v F(!t
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paioAd I, titn
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s1"1la
rist
F(HJt F[t) 't
+ '
t" t!* <rtoJ t+fliA Lt += u+T, i. .d,'d'rrJ lrrr#r{.al
Jr? ;r(.*rr 1n*r)Ju +
J.
r-st'-'D
s(u+er)Ju
tu ;"" s1u1l't
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T >o
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o<t <r.
n <t <2tr
F(r)dt
ftnl tl* L"ptrce
rG)=4
t
Ftt) Lan P{rtuJ
1i nttrj=
rl
,-,-'*"I
1r
:- 4ns- Il-e
L
r,r
t-;*" IL
F,..?r.'
a- .'5'l
h'4 ?Bt6l T=atr
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eir rLe1t
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J"
F(t)
I
' Z1F(.t -c
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ljl
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e-tt (. ssl"t
- c4 t)
+o
t:Q +1
s"+I
Q-d")GtD
, o<+1L
F(t). ilrn
.rJ ect.t"
F (1+t) =
.# rr,"-rr,.,
-(.{ta."2) €'3 +z
I,tt' 4 L"rl"e )r^t""; {.t so* 4lc^-l Jnh.H,s;
f (+)=irtJiri {, IFE =i(t
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@t
g)
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^f
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f 1i Ftrl
t'"$'- d
lnverse Lallcce T"^*"*
.'_-.-'--
r{ f(,) i, d," tael"cz+"a'J"ta $'$"t"- nrt) '
-=
tG) , t:,Ln FG) it c^!d 1t'a inwa' La?t^ce
{e Jurctfdn!6 ^l
h 51-'"Atd ''t;Lrea
Gt n"e"rlt4 rc"erb;'
rY'f-
n"L:'ltct
aw" L' i, c,Jtd X,e ti'"*,". t"4.* rr^$r-'h- ape"&or'
a+ptrEa
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arc ar, too 6an^'& +1"
"
. L' 1"'
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u'* '.'J'c'r]='a'1;'{{'t'r+
Ls) Fk5t.+'4nl^.h-(' *J+Y P;.4bJt-
tJ 1;'{f(t} ' Ftt).'ttr-
x'{f c'oJ= 2"tva)
9t 14" 4 La?rac'+'rir!'-Fr-ae ra&
jo. zirro]=Jlr'Lnt-rle'
Jt'"o ' Jj
-ts'a)rFteet
' J, u" I e"trtttlLt
= zi"or9
.Tllrc'-t=
'*Lor<n'.-
eofFto
7a'pe.+4,-'
_t-t
,t....
r'ir"t,rl = t {$lr.,r}
=(-r)'drttt
on1, n =4,2.3,-.---
Laplacetat*fun { ;*nn Lct-
I{ J.'{frql = F(t) , tr'{-
t'{f"rtoa'1.'=f( _l
rj -{:'iict=
a-'i d" fc)l
FG)
* {f(tl "
F(t) , llqr
5z&"1 +k'y,latid (ar srSiti'6 ni,r..b) .-
&tr) , r.tL/e Gtt)
r({-") )4
rJ *'lf(ol"
a'{ fatl =
Gjr 1""-* tataeTr^ufua{
ij
i
Mu,ltrlic,itn ld sr ;
{ {''it(o.} = ftt) 4'"-l
.I,'
| :'lc":J=
J-r {
-
rt
J"
.L
-
.),
.- 4
rn o;ni';-u; s:-
rJ 1.1i(t
d'
.MGY$:'
t! 8d
= F(t)
{ LE{TI 'l
cl'l
I t(f'*4) J
l,J:t*m Ot t^aue'
= t FLu)d4
J".
- lorv) l-
'
'l 3in2t - F/+)
"s-ni./=-T''
3:!+ 14
| - t-c'sz-l'
,.1
t I - c"sa'_.1
nt
ry1
J-r I | '1
a"'l@aJ
= (t - cnau)J.i
-li- -L sire'!
6
l-t
rL
+t
=
:
.t
t-
I
l4t
3i
i
s3(s"+91
+ r'{rc,rf=x-,ii"*?=
")
,]
+
= {"r+"-*
= [];"iz
= t lt -6) + J-11
= _Tr,* i,,-
--tS
S.t
' r 6"-,-r",
t-4 s,"t"J,
1,'i
,z^",
=tr F!t),
'iL "!f
Let t,s) . -L<-4.d-= j4f(91
= F&) -'- 'o
t? tLad
L.t F(t)
F xb 'A
6[r) "d-too fl'cn'o-t F(tl aJ
u6 'i"z rzl#ior
a =
Jt
rtuatt-'))'r
=
J! ao p(+'qJu
.r'6 v. dtD qllzl t't
4 +r"
J'Lf"'?)
ntA /zlJln f.
$ o 'ta
0 F*c'
O rxc''
Qi) F,e er
l.sali4hf ot tdLTar,?
U
b l;<,hiLolve
;,e, F*(6+
;,", g*6= 6*F
;,",(*rs) +u '
F* (G'u-H)
a.r.to aL/,iha .
il) = F x6 + F a H '
s6""*^^ -r.eatn:'
.tI
-.J a-'|to+a)*JJ:'It|oJ
-1:'1JG)tE-,f 't"
*
=
Ia 6,u)
" GL+), +J"'n*
G-q au
FG -u)/a
s),
3
"dstr1pJf
--
^e'G('d'ra,te66@Ld Jl
e-'rJ(,)G(!)J
J { r Ct-it,(t-t)?
L ')
| ;" r G-) u(t-t)e
't" G[!)-F(t-])
f i'
ll
"0 J,6?
'1
jcs),fo I
=J:
,.J"
lri"d L.t
=!,''
ta
,.s)
l(
,
(
c
G](
P4'
Jc
d(
:s)dG-t
z''I""tr"t- Fu-9uG-9)..-.@
e(lv., d
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(+- ) k w;tttL?fu.tli.n !t^vx;//t u* fuoe'
o , o<t<f
clilarl' r tt-')*t'''t
a >r3
b(I of <""o tA,i
'fcq =
-'f<': =
@,
_t-t
l t:il U I t--
1(t:ll
aLc
) dt dJ
F{ft @ 4r @. o
dt tl
3=,
t"hd(.i n d".etl-
r"-d.l l%f t =J t
1r.4 (lalel
l1ro
.
+. = "4 ;) a}4^ !t4
f":=
e".6r^4rtz orrer
&6 3=o t. J=-E
toats)= t
?s" ("() F ['t - J, 4J 4
'
J,' I J"
[r- 5) dT
aJ-'{ft,raoi.
=J { (tc^(t)F(t-l)
Jl.u'F(t-1)iJ
t'tr"1rrJ=aF(t--)u a)
d ri%.{4h'", r
o. AJ--
J.j
,,J
,tl
a(t) F
124,
E",Jqtlt z-'1.*"!r-],u.* f
orila s s- _)_
sd
Ly+.')-
Fri
*-.n *-t 6co=frp
'{f(rJ=fl*-*,J="-^r
'ia."rJ=r-1+"1=t*
H
(t)=j
rt) =l
.Jlnca p
e4 cI
.1"
- c't+t s;ne)d.u .
ft
J6
.th du .44u dlt
L^c,'"t'(t anuu dtt .
e-"t f c^t*lE-2?
L
----z!:) o
"!rn -tt! c.'v.tqidi t@,e, _
+cstr'| =
f ru;G(+-qe,(
T = lr "*",.
tina(+-u)
rla
r rt= -F
).
w* ('ih^+ c,saa
=atr,r= +it,n )6c4-^"
lu -.-t c4@+
=h ,,."tj"t(,* *,o, lo _
--'Bf, * s;"z"qlt *44 L 2,^ )6
1t^-'**,%
1,i
Y-, ( I
* 1fl*,9r
'4*- ri^t [1* ilYa!
' -;-a- I ' 2t1
(c^z^t - t)
-ff ltst;"'iLc"s+f
; ,-if -
+
sinot 2rinal
'-.:
'lrotu)(.
-+)
f
0,t7,'f 13"+ze"+I
sX @i{A &ny.tltid
fr!
l7I-*oj
rrl
I f,r5-+9J,
,t -2
{(s"
4
./th!4e,-*-r--
. r{ ,ari)
q'"G,.)
h@ n di{nhcf
?n6.'
sr."" ?[s)rz a pldnm;el6 de6. k ,41t^h it"f { qL')
a,{ q.(r) hM dietin.t eeros {, 44,,,.:.., dh , oe 4h
oritc asq6irf;*L fit maiLJ
S f"'Aa fuh:"u -
K'
Y:)
(@t
e
QCt J-rr
uull''llyr.6uftf
p, c,-"1=(s-/*)
.an
J-{n
An(
'-"t").. * Ax +- .
":l_xn
6iln
Ad:!)"
-r-, 1P(s)/ _
Y { ao)) -
stde'nent;
rzt Pcs) 4.1 Qlt be
aft,1""4 i:@< anil@j'qt,)
ir4rdsdx, 1=r,r,. .,n _tl.-f
E(') = (1-d,) (r- .1,)
rA
L PC'(")
=392 .++oPu,t*+...
' {l4i) d(dt
-fl,sn
]
-Iaxi[ L' H,"?i+^li' 'rqle,a.t@
' '#o'
,' l
?lu") *1, -7r'
sJ,(( A.l(t
f(dd
a,'t4't
o"-!9^ de,
I e+t-1
,1
L
8b t H63Pii4'r Yurl
t(e")
d(4"1
f.r"^.O a er.t;t J_
--@t,, tt*q .Y ,
Plf
i
1 )r
AAj s-44
-q{r)
=
A:'(o
t-. o?:r r-"2
2 t-<11^,u 4 O
-"I'l-:u'fl"-:,.))"-'^'-",r^o":"'-,:.,?
rr"-4 |
G+,C-a)(s-, l
5.tn Q+r)G-2)Cs-3)
8+,)(s"-rs+t,
s9- 5s'+ (s +s-- 5 s
53- 4.,--l-F + L
t-,1
?(5)
- -t, a-' -e s-z I s9
Ub Jleav;sil"'se<ya,ai."fi'-u'a -
z-{4*.?, -+i+,1-1-r'[+1+1r +"1I. q(r/) 3 - {s-2 z^ ls-t)
Jr f.ff=G,/'-r1- -tar-3 021. !".t. |- |
i"r' " -,
2+t t_
2,
i.
Eva-"^tt
4-'.1
t
Jrfl:a,w f
Lafl"ee+n^4.4i,r^sk EJ.arlial .fid-,-s
(A)A?pli.d,'n{f Lad"'€
d;{fardb'a.le4'{d'"s o;in
'tr4nCrn ; sJ'ia6 orlr"aq 'L'aa"
c'ulatt cr-".ff
iz"ts:-
O s"tv.+r"'JillddrY/r$"'tt4l ery"li ud uliaAL^C...-te'5Ji'":
=
f.*tr*t
y"az1'. s'Y= drsi't
l(o) = o
Ytot't
sln :1" 6i*n )$ln.'ri e$hq t'5
y,,+2y,+sy - nts;t ....O
-r^kry ij,. L.da. tdh'J-- 4 btttaitaa4a ai |,,t''trx'! nr'€ncnlillzr
""
:ir;t n?11'{'l* s'tlvi' {{afs;'i,
s"u{,)- sljq- tt,)*lJsbG)-IgJ+
'd(s')
{:
J+
det- 1+2riri)+s[@.- @fit
ic.+:.*s)'l(.r
= r +irifir
_--,L,-- + I
Q--+
zs+5)(*.a"*1
dG)= tr+t
2t+9
')({**.s
@
t
s as+2 As+9 Cs+D
+dts +t
6", r:)(r^ir'-4 1(cs+r)(s'+zs+:)
fi++c) s31 (2116116* l) s"+
Q,a
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at,9t+2s+2 =
erdT
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5A+nB+eC+4)=
59+ 2D = 3
+'ra Lll.ho'/ 9r A, 3,.
su+As+2
2
3
&1r';,.d
5e12s+ 3
z
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a
I
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lace+na,vfura,
6D=rJ - 3='
3- a'
'J c;itrt
Gxt^6 +e inwr* Li
t;'l-1ro= 11-'!aI
Y=YlO=*ettl"t
Y " jtt(si,rt+ sn
=)
;
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rl
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:a'i"r* *, J,rr,"r,'0$,$,'.;hr alq,i"a !d u!i'8 t4t^4 +@tJt':-
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(o)= 1' t@=a, rt(")= -+
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t"r- 11"+g'r'-Y
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- i+ 2-3 id+ls+ 33d-3-b =
G53
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) r-'{,}
t +1.;"'. ,{i'
;'. ri1'
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€-r'
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a
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ft-rf-(s''1-1 .2
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Laplace transfrom lec

  • 1. %?p F (t) !z 4 tltahsJlri 5 ' JlPt'rJ= tte 'r1"c" Q) cnu"u. fr .s"9., ')."lu.p {{ sl3,'r &wtqvs_'. +"""h'r" .l + 3e..i$brf* 1>o . zlta F{t) , denltul,{, +1rn'1,tt r9Jtnd6d- .....(, ;,s.sail1oe$st y t* t*X'l ; d+?r.!isc it d6e4 ntt 4*i3r' ,} i ,I pnoviJe!.ad.J-tt{ ri+erdt (1) r:ish. 's i, a e<t"-&.. at"t 3 te reA .r c'nplix nu-ler. ,rolL.ir..":+;;'$,^ 4 t) YIrl* i'l LaflaceT;urs$ro fle)^i'rt1 sd rd"it' { tad.a +tuhs},^."az h"a
  • 2. r.-:, 0DF"d7l+i " 'srln 0;rJr{l .f fau* t't "3"- ' oe t4'e-_UJJ ' J,{t1 = ({ o-'tr at =4. (Pn-.tt et =+ s'>" Laplae tc""s$r- e :- ,4r,.dG.--" $b.ri'l {fi ,{ (jD'r',.flr,,r *nd f,{ ]. ,. ,l,,fqr."1' I,[if' | *i'A t ) Jr =*l,c " -:i
  • 4. (vDF-r tr { &s'i} ' ":1t e6 t"y.r+'- $ c"sef,jt e-st (- s c" {,l c"s't| ' JA D- l' h-l /;- : +-t,vf,,"'4-i-J
  • 5. P,r"y"ra4 L^ru*n^"*'. Lr Lall"cerldd;" l, g ^.. ^nd o"rl?ant a{tb L,{ac"ln"ia;ft"- J,Gl *l I {n, r,ttl-'u, ,",s"'^ * .ra1lrrta] . egg&: Fina rj. ,+rpUl^6 i"1'*{ (,at} e L * :1,! i; stl o 5. (9) sft3t tn-d^A'Y'(d sf) -n^--,. gf 1{ f (4=Jct . a I"& F,lri'l ' ! L' .., e""r"h ii-th = J6r-4) b'
  • 6. - --ri ^d+ FLt)Jt BS- "","n F(t)AL T J; !G qa{^9t€:. = J@ (5.4) a) ,t )a 1<4. ) .ll a (r*s1: :) s-*.ns t r","lJ- C- "i,+*; ptt?k'.- n4feh:- {: 1lr(t)l.=JG) -4 c(9 = 4 F(i- I q., JJ-" f, {6ttt = t's^ft"t pndfi- 5e reve - - 1 1GLr)J= [i .t't.au>,at. = .f3;'t atttat *J" ;'t ato:+ = [i "'tO:at + J" ;'t nG," o + !" lt F(t-ddt .--- 6 +i"uF.- d? zl""+rto]= d€ h4* 11t3J . 3l sr L t tJ- 5l L Ld ur^d idl- ptuiuZ ! at &ve - 1{ e'Fti} =
  • 7. Let t- a : u.,." -2t: -It^'A !'a A ,6? k^rz- -1'{ 6Lt8 O) c.nde r= a] t=' I!r:a) - .. at.. &- F(,rt=*j(*) = f" ;4'*D ,,u1su J. -, 6saS' .5u F,U) au = 2 t)Je) . { sd€ Pn+."fu'.- Y{ L{ F(ol=f (") , lz--or,l , ra -(1)" +J(*). Ft4{{ = d*"1{ BC' ou "^"..-1, {F(^t)] = |"."-st Frq)Aa ..o Tr^^nln O. o. t',." - -rS 7
  • 8. qr,"d.h"C.,€fs..r" S c"s"t p""r".t , .f.r {.4 L^pr.- J"+1 ft*l;' - 1,! c.:t] =JG) s +!(+)r1.ic..^tl= -L 3/^ J.,l'4 a lu^r $ IIFOI. (rs), +z,v zlr"r+tJ. - sF(0 Pv&- '0 ,,t p1oJt 1;frad- f' ^ --Le Ftt/arFrdl+sl to) -f sJ(5, - F(,) . ddr 4 L"?ncz ri;q.{ = sltO ffc.) (F) t-.plq.. 1*"$'r d j :f 'a';. {ro ,tL"-l,in?oJ--sf(o-F1") Pn',xft-bl L.gcc 1n""*f'n. ae t".t- 1iF(Dj. Jn = drt n(rttt e tt r 'ct):t r J: -Jt " tlt - e'o). - t(')
  • 9. x.1r"'(E= "io- '' -1"i F"Lt/| ott r"(t):t 'P-'*r'c]ll.' =a-F'u)+s 5 &"hn"i.6 thc P't'ce)o' 6e c'^ - {-' r(o)- s"'" c l"o) o (d L4pr^cei."J"- d h+'Arn):- 19:3- " g Tlsc)l=JG) ti4" 1iJ: rG)5 FLu)1."Lo.+ &(t) r'"." d (t) c'(o) lF(ol-er?6)- r'?') F'6) - Ftt) r,,o)_.-.-f"-'La) =J: J."ttot'a' S'e'rF'OAt 1| rtu| l,sc"r- rt"l] (,) - F'(o) . = - r'co)n ={!G)-s ]- J: ,'t*t leto a'o]= - 5"-l FLqduJ = 1 I F'd)l P,6. fl eia N'o .1{ a L {ru>}' , s'Jc) slu)fu =a I lt rs;kl {rs) 5 s1.{ {t rt"r&l l{l'F(u)dqf' tvo
  • 10. i i./er",a4 t^l^",ti^"rJi,.( M,r,'dr.]i"h ft o*j9: - $1,!Ftt]'J(t, #.'j r if rcu]=err"S"rc"r=r,j,.:,:= ,.,. *J! rt ecD -r,4' t Lerrhl+!,s #"rto (G) reri ,6. lle Aar d* F[0..dt=1"fo.r t", Atfeh^h',A,^d.il.rxr t"+ftdr6{^. Jt rule f' n=x tt4 I " J"ft ;"'ornat = f'- t;'trto dt Jo = - [' a'titrrb]J" r " --Jjtnttti' a a r.-,..1- )t '^(, L 11l(r)[ = -:r - _JG).. DhicJ. pr|do, {! -{{oro- &r n = 1 -n 6{^tris!, tL, rL-,.- i. i,v,a., . o€ lnl(ciri .1rE(--l't {lat 1ir '{r.uvea J,"*i-: (s) -@ '.-.1 FoJdf
  • 11. -rr^ +(o ust;r" pldJt = , -J:t', lt..' F(e]rt > fru,r t"-' rtO]lt .J" €d jG-',1 =tf 1t*u,, " CnJ-'Jq'tctr tr rt tf((e gllrost'J S O '.' +'* '" d1t- ia +",," r,' +e fleora t"'1lr r-" A tU ll^ow tt ku lt 1t " n: {+r =2 @vr 't" 1,.zTi^6..' u +""a +' 'U p*a -1,t t' "*+J c4.xl = .r--_+4- r-,P-1lE- r-. ts. LJ-i47 "13 _ 2ts -@* r;s +tu*.^ l"1/,l$' . =t1 &' n =x+'l .+L Jl-;' il h 3 *J s'- p"silv" i^1"4t) vJuto =!@ (t4)
  • 12. A^.F(t)1 - r! =J -.1"'d" T (11) l;"ist"n {Fc)l ={(s) -F(9t tG et JG) &tr) F(r) 11F( to - JG) s fn,- s t' ,a, - I +(r) dt J3 -- l +Cr)dt JlctUF .l {r)/r I 3r5 {c,(Dil') E4 L"t tr) It *.tol *1{Grej oJ=) ) ) 4l icrol l*.da"i'd t g sita d.r.+a Ltr',rJ:= f+ 9ia .li61I{o'to1 u?1oc.1=3 11s*^ 0o= o !$) l!ott +tikl,I =f '),t-Ltfrt,t.?t,rl' teJ = | foat =f'oJ' = ), t t";da -) )z
  • 13. e'tJ] r.f rl9 r- X{ettt]= f" +T) 1a*. rJ Oe J] {i+ {T flr)t = I =it -!, -T)o F(o2t $ro !d 1l F + r) 6e FI t ?T-st * )","'lp(oJt+f{ d4!"tk" 4 ,t=l: =1" Ftt) otz. Etl a3uF$lS+ i'v F(!t F(r t.pl".c trana$'"., e-stFLt)lt paioAd I, titn )l' s1"1la rist F(HJt F[t) 't + ' t" t!* <rtoJ t+fliA Lt += u+T, i. .d,'d'rrJ lrrr#r{.al Jr? ;r(.*rr 1n*r)Ju + J. r-st'-'D s(u+er)Ju tu ;"" s1u1l't r+".'% '6*. -=;: ,l"c - t*'l:-sTfT +e I e".'r] -t:;n Ir'" J];'* ntort --r'fi tb
  • 14. T >o I e-' o<t <r. n <t <2tr F(r)dt ftnl tl* L"ptrce rG)=4 t Ftt) Lan P{rtuJ 1i nttrj= rl ,-,-'*"I 1r :- 4ns- Il-e L r,r t-;*" IL F,..?r.' a- .'5'l h'4 ?Bt6l T=atr J" eir rLe1t (1".'+.,rttll++[' J" F(t) I ' Z1F(.t -c o;'trtn) ljl ( '' e+] { rtDf . ."j".'i.'rritJt e-tt (. ssl"t - c4 t) +o t:Q +1 s"+I Q-d")GtD , o<+1L F(t). ilrn .rJ ect.t" F (1+t) = .# rr,"-rr,., -(.{ta."2) €'3 +z
  • 15. I,tt' 4 L"rl"e )r^t""; {.t so* 4lc^-l Jnh.H,s; f (+)=irtJiri {, IFE =i(t At @t g) "^1: @ c-.( ^f @ t siha+- (D t c^a*I @ e"+ ";i (!) e'r c-sst @ en+s;,r ti @ "*; ; .Jn (f;t" (9 t't4aL+
  • 16. Ddn,- f 1i Ftrl t'"$'- d lnverse Lallcce T"^*"* .'_-.-'-- r{ f(,) i, d," tael"cz+"a'J"ta $'$"t"- nrt) ' -= tG) , t:,Ln FG) it c^!d 1t'a inwa' La?t^ce {e Jurctfdn!6 ^l h 51-'"Atd ''t;Lrea Gt n"e"rlt4 rc"erb;' rY'f- n"L:'ltct aw" L' i, c,Jtd X,e ti'"*,". t"4.* rr^$r-'h- ape"&or' a+ptrEa , r r., ^& a, 'dz, 3gs"' rJ 1,1r,tol'f,tt *J l{r'ct arc ar, too 6an^'& +1" " . L' 1"' '' u'* '.'J'c'r]='a'1;'{{'t'r+ Ls) Fk5t.+'4nl^.h-(' *J+Y P;.4bJt- tJ 1;'{f(t} ' Ftt).'ttr- x'{f c'oJ= 2"tva) 9t 14" 4 La?rac'+'rir!'-Fr-ae ra& jo. zirro]=Jlr'Lnt-rle' Jt'"o ' Jj -ts'a)rFteet ' J, u" I e"trtttlLt
  • 17. = zi"or9 .Tllrc'-t= '*Lor<n'.- eofFto 7a'pe.+4,-' _t-t ,t.... r'ir"t,rl = t {$lr.,r} =(-r)'drttt on1, n =4,2.3,-.--- Laplacetat*fun { ;*nn Lct- I{ J.'{frql = F(t) , tr'{- t'{f"rtoa'1.'=f( _l rj -{:'iict= a-'i d" fc)l FG) * {f(tl " F(t) , llqr 5z&"1 +k'y,latid (ar srSiti'6 ni,r..b) .- &tr) , r.tL/e Gtt) r({-") )4 rJ *'lf(ol" a'{ fatl = Gjr 1""-* tataeTr^ufua{ ij i
  • 18. Mu,ltrlic,itn ld sr ; { {''it(o.} = ftt) 4'"-l .I,' | :'lc":J= J-r { - rt J" .L - .), .- 4 rn o;ni';-u; s:- rJ 1.1i(t d' .MGY$:' t! 8d = F(t) { LE{TI 'l cl'l I t(f'*4) J l,J:t*m Ot t^aue' = t FLu)d4 J". - lorv) l- ' 'l 3in2t - F/+) "s-ni./=-T'' 3:!+ 14 | - t-c'sz-l' ,.1 t I - c"sa'_.1 nt ry1
  • 19. J-r I | '1 a"'l@aJ = (t - cnau)J.i -li- -L sire'! 6 l-t rL +t = : .t t- I l4t 3i i s3(s"+91 + r'{rc,rf=x-,ii"*?= ") ,] + = {"r+"-* = [];"iz = t lt -6) + J-11 = _Tr,* i,,- --tS S.t ' r 6"-,-r", t-4 s,"t"J, 1,'i ,z^", =tr F!t), 'iL "!f Let t,s) . -L<-4.d-= j4f(91 = F&) -'- 'o t? tLad
  • 20. L.t F(t) F xb 'A 6[r) "d-too fl'cn'o-t F(tl aJ u6 'i"z rzl#ior a = Jt rtuatt-'))'r = J! ao p(+'qJu .r'6 v. dtD qllzl t't 4 +r" J'Lf"'?) ntA /zlJln f. $ o 'ta 0 F*c' O rxc'' Qi) F,e er l.sali4hf ot tdLTar,? U b l;<,hiLolve ;,e, F*(6+ ;,", g*6= 6*F ;,",(*rs) +u ' F* (G'u-H) a.r.to aL/,iha . il) = F x6 + F a H ' s6""*^^ -r.eatn:' .tI -.J a-'|to+a)*JJ:'It|oJ -1:'1JG)tE-,f 't" * = Ia 6,u) " GL+), +J"'n* G-q au FG -u)/a
  • 21. s), 3 "dstr1pJf -- ^e'G('d'ra,te66@Ld Jl e-'rJ(,)G(!)J J { r Ct-it,(t-t)? L ') | ;" r G-) u(t-t)e 't" G[!)-F(t-]) f i' ll "0 J,6? '1 jcs),fo I =J: ,.J" lri"d L.t =!,'' ta ,.s) l( , ( c G]( P4' Jc d( :s)dG-t z''I""tr"t- Fu-9uG-9)..-.@ e(lv., d " (+- ) k w;tttL?fu.tli.n !t^vx;//t u* fuoe' o , o<t<f clilarl' r tt-')*t'''t a >r3 b(I of <""o tA,i 'fcq = -'f<': = @, _t-t l t:il U I t-- 1(t:ll aLc ) dt dJ F{ft @ 4r @. o dt tl
  • 22. 3=, t"hd(.i n d".etl- r"-d.l l%f t =J t 1r.4 (lalel l1ro . +. = "4 ;) a}4^ !t4 f":= e".6r^4rtz orrer &6 3=o t. J=-E toats)= t ?s" ("() F ['t - J, 4J 4 ' J,' I J" [r- 5) dT aJ-'{ft,raoi. =J { (tc^(t)F(t-l) Jl.u'F(t-1)iJ t'tr"1rrJ=aF(t--)u a) d ri%.{4h'", r o. AJ-- J.j ,,J ,tl a(t) F 124,
  • 23. E",Jqtlt z-'1.*"!r-],u.* f orila s s- _)_ sd Ly+.')- Fri *-.n *-t 6co=frp '{f(rJ=fl*-*,J="-^r 'ia."rJ=r-1+"1=t* H (t)=j rt) =l .Jlnca p e4 cI .1" - c't+t s;ne)d.u . ft J6 .th du .44u dlt L^c,'"t'(t anuu dtt . e-"t f c^t*lE-2? L ----z!:) o "!rn -tt! c.'v.tqidi t@,e, _ +cstr'| = f ru;G(+-qe,( T = lr "*",. tina(+-u) rla r rt= -F ). w* ('ih^+ c,saa =atr,r= +it,n )6c4-^" lu -.-t c4@+ =h ,,."tj"t(,* *,o, lo _ --'Bf, * s;"z"qlt *44 L 2,^ )6 1t^-'**,% 1,i Y-, ( I * 1fl*,9r
  • 24. '4*- ri^t [1* ilYa! ' -;-a- I ' 2t1 (c^z^t - t) -ff ltst;"'iLc"s+f ; ,-if - + sinot 2rinal '-.: 'lrotu)(. -+) f 0,t7,'f 13"+ze"+I sX @i{A &ny.tltid fr! l7I-*oj rrl I f,r5-+9J, ,t -2 {(s" 4
  • 25. ./th!4e,-*-r-- . r{ ,ari) q'"G,.) h@ n di{nhcf ?n6.' sr."" ?[s)rz a pldnm;el6 de6. k ,41t^h it"f { qL') a,{ q.(r) hM dietin.t eeros {, 44,,,.:.., dh , oe 4h oritc asq6irf;*L fit maiLJ S f"'Aa fuh:"u - K' Y:) (@t e QCt J-rr uull''llyr.6uftf p, c,-"1=(s-/*) .an J-{n An( '-"t").. * Ax +- . ":l_xn 6iln Ad:!)" -r-, 1P(s)/ _ Y { ao)) - stde'nent; rzt Pcs) 4.1 Qlt be aft,1""4 i:@< anil@j'qt,) ir4rdsdx, 1=r,r,. .,n _tl.-f E(') = (1-d,) (r- .1,) rA L PC'(") =392 .++oPu,t*+... ' {l4i) d(dt -fl,sn
  • 26. ] -Iaxi[ L' H,"?i+^li' 'rqle,a.t@ ' '#o' ,' l ?lu") *1, -7r' sJ,(( A.l(t f(dd a,'t4't o"-!9^ de, I e+t-1 ,1 L 8b t H63Pii4'r Yurl t(e") d(4"1 f.r"^.O a er.t;t J_ --@t,, tt*q .Y , Plf i 1 )r AAj s-44 -q{r) = A:'(o t-. o?:r r-"2 2 t-<11^,u 4 O -"I'l-:u'fl"-:,.))"-'^'-",r^o":"'-,:.,?
  • 27. rr"-4 | G+,C-a)(s-, l 5.tn Q+r)G-2)Cs-3) 8+,)(s"-rs+t, s9- 5s'+ (s +s-- 5 s 53- 4.,--l-F + L t-,1 ?(5) - -t, a-' -e s-z I s9 Ub Jleav;sil"'se<ya,ai."fi'-u'a - z-{4*.?, -+i+,1-1-r'[+1+1r +"1I. q(r/) 3 - {s-2 z^ ls-t) Jr f.ff=G,/'-r1- -tar-3 021. !".t. |- | i"r' " -, 2+t t_ 2, i. Eva-"^tt 4-'.1 t
  • 28. Jrfl:a,w f Lafl"ee+n^4.4i,r^sk EJ.arlial .fid-,-s (A)A?pli.d,'n{f Lad"'€ d;{fardb'a.le4'{d'"s o;in 'tr4nCrn ; sJ'ia6 orlr"aq 'L'aa" c'ulatt cr-".ff iz"ts:- O s"tv.+r"'JillddrY/r$"'tt4l ery"li ud uliaAL^C...-te'5Ji'": = f.*tr*t y"az1'. s'Y= drsi't l(o) = o Ytot't sln :1" 6i*n )$ln.'ri e$hq t'5 y,,+2y,+sy - nts;t ....O -r^kry ij,. L.da. tdh'J-- 4 btttaitaa4a ai |,,t''trx'! nr'€ncnlillzr "" :ir;t n?11'{'l* s'tlvi' {{afs;'i, s"u{,)- sljq- tt,)*lJsbG)-IgJ+ 'd(s') {: J+ det- 1+2riri)+s[@.- @fit ic.+:.*s)'l(.r = r +irifir _--,L,-- + I Q--+ zs+5)(*.a"*1 dG)= tr+t 2t+9 ')({**.s @
  • 29. t s as+2 As+9 Cs+D +dts +t 6", r:)(r^ir'-4 1(cs+r)(s'+zs+:) fi++c) s31 (2116116* l) s"+ Q,a +ze+:ctzr)s +r o+: o Q"+xs+fxf+2s+2) at,9t+2s+2 = erdT Jn"mt rii +hac"-.fiae* 5i s', s",s *'A n' c^*^+ i'"-') e;le/,, .re. 34,1- A+a-c,, qA+B+2e+ 5A+nB+eC+4)= 59+ 2D = 3 +'ra Lll.ho'/ 9r A, 3,. su+As+2 2 3 &1r';,.d 5e12s+ 3 z t,t dL Azf 4e G Yfr) lz S-+lsfs '+2s +5)({+2s+4 ln s"+2s+2 a I ' i Q+r"+2, lace+na,vfura, 6D=rJ - 3=' 3- a' 'J c;itrt Gxt^6 +e inwr* Li t;'l-1ro= 11-'!aI Y=YlO=*ettl"t Y " jtt(si,rt+ sn =) ; " ,+l rl (s+')-+r"J
  • 30. :a'i"r* *, J,rr,"r,'0$,$,'.;hr alq,i"a !d u!i'8 t4t^4 +@tJt':- .ra_u.rr *21t-1 .lJet. (o)= 1' t@=a, rt(")= -+ {n * 6,,". J,g-"-i..! e"-t- "+* t"r- 11"+g'r'-Y I?l.'.i{, ;;* ' +'^*5'- "3"n "u* d$o eJ ,srt tu 6t'€L c-riH"rs- o. 4 .,y, ^-,,,t _l11 --7-{t'"-t . a{Y"f -311Y'i+"rrrl -' -'.i,-al.r-.rh,-1i"r-rr, Jtr-'Y{')-1i'}} . .i ! t-r-l" 4-'+ i -- +31,.6-ytdrl-6 ' r-ttft-l;;r - i+ 2-3 id+ls+ 33d-3-b = G53 -0 ) r-'{,} t +1.;"'. ,{i' ;'. ri1' -ts"*s-)X = t-3s+s" €-r' r-35.rs +G-3 a + G;tt ft-rf-(s''1-1 .2 ,2 .;+ .;iF".,;-'H;l.'*'ito"l ut - 1J - 5t"et + -.t-*t5af .ji I 3e
  • 31. LB) Appric4lr'-4 L^p144'f,ti6i/^ JStrea$J Euah'4, oi eariatl€ ,t t"6r..tt ,JA ^l sryr,:d orar&? ti.,,,ar c"-e.S'.ia,-dE. & sdye {r Srl"or^Ad#ra!r,J a1"4.nrd.d'U r^a 141"'L t'^b{FrF:- Y"- tY'+Y = t Y(o)= I '1'o)--2 4 n, 3r^"Y$i."*+r^l --- y'/ - t Y' +'f '1-o,r<t'nu.pl*.t,^"df.' U^li+itnt ' x. ?Ll z{v'J- l,ltr'i +71rJ- :rfli a n-nr';-v1q++s1"U-yid]+y"* =) dl-s-a+s{+X-o +tr= ! , 5U'+G'+2)T= s+?+t i =) x' +l)x = i+?+3a ) bu *(+!)7= t+$"! -"(') dr'i.h ir - ft* rJu d"tl. 4t. r.F ' nl',)'s " "{""h'= <t't" . el-ai" = 9 "{t' o *rZ
  • 32. {"*'X= Jl'" +r+)*"'"''a' il =+ un'Je.*".-n;"o"'' = " ,',,1r,''ut,,,1,";,.t,,r Ji1:, ' U"t'iI tr"za"' otu-"iu-Ji''"'''f =+ ut"l! r"*'t,*r,"*'*,fl--Jn"n"r'l +-Y --.1 1traQ't'.lin6@"- -t*,fr
  • 33. )ihdro.neMr orath(q .Sr*t. Foqahdnj. 5oo< Jy = aX-3Y o JY Y-Xx Jt sqLjlrt f. x(o) :8, Y[o)'3 5i!' 6e A,4;"l1dL,+l^.et*vj"- o t [il, * *14|{a"-"',1 s,1"i1@' k @ s;"""t+"^.^^tn., , = i5 :,] - ',-,*-s-z 3 ] {-b-A 2 3 t) s-z e 3s-t2 r=1;i;="'-:'X=/'(,{}.?.:'*""'t I =r'fui = sat- t.^t ^iha 6 a 'uJ @) =) s,,(- x(o)= eYill - 3JJYl - S7-8 = z^-Bt ) (s-2)'(+3t=e .---'0) 71#t =:lv'i-"r{"1=) st-Y(D) =1-j.7 --)5[-3 = Z-2x =) qn+ (s-,)7 =g - @ 3
  • 34. FXar.4is4 @ ntc .lt l,.l.'* n= t ,, l,(ol.''1,I - ftnrl Y= t tc4t- €'i"t s;lvc '{ v,r + 9Y -1 hi'e Y'-Y tY(o)' ", 6a""^'s 9";,) ..i
  • 35. ,F T+ U fu) , -114 Ftvc-r9,F 4ft9 - lr PP)- t'' r|i - ' -5F lo) Ilr(D]: 7,tF4(tl= - r rratt)'- flef$@ *1 -l- l.tl ."[, _Frg W a;," r*) TlrfuJaf<l 6e>,'j'tpnn'rt'!l'oir6*:T :!*-*:.{,r;!fr, . -",,n - s'.(o) - s"-'6(o)- 1-'F'1") - "'-tr (6J-r t!1,. aFw1 - -' 'l'1 Aw rt l-+; 644 *nU'- loLifir^ ' aetlr'L - . ( -rr._/94)_t L F't+11-- )t " ' ,T " ga os+5ir-))1 -- k+tF(9ja * 5J" ' = o-e"rts)*sxirttlJ = sftD - FG) ' Jcr- -ttr ilr't'^n ;' +*t Jot n' 4" tt p- 14dk L 5 F' l I -e + = a 0 t t ( ( L' t I s ,lt )o oJ"+1 F^(, 1 " j r"'', 'iy 4'b f,"' {r'rolL t t { r"''a;l -l t] ()' -t.
  • 36. ! 3 F Lo)+jf ll€*'(DJ= -r-L -h-l-tt !)-r o) 9 F.f"t +tr I f"' (D]' '^"io - { Fr")-.s'" ri}) --, r.'(o)(o)- -' 2"t-tr--1,/ m2 7 ! r"to1 = {'il'r't' {r(')-{^ tt';- -'r;;f-'ret if tt^- lt.'' ;s *^" !" n" ^ ' 19c'* iF ir +'''{{ pn "4g.X fr. 1t""'/u' 44" {{tt) - t'-' Fto)- {-2 F'La) - - sr"-'(o) t -!ln'(-rt ,R , 4' L+l = 4 _r"-r(t Io. n' rl'2 ga-ore- i5 trd F "' 9+' " e'+<^ #,a-"* +tr' +t-"ftP il I u-z V J e"caftt'l t^{..(,.,94 , i4"rleLt]"