Laplace

December 2019
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!"#$%&'()*+,/ 0!"1# 2 $ 3%&4/ 2'"56($ / 7#) .

89:

Laplace

;!"#$<=%&'"(>) * ? +#@$ A,-./01- &2/3>4 5B 6C?DA1E7 A

FGB,=HIJK ? LMN 89:O89:89 : Q%; P f : [a, +∞) → R R!* S T UVWXQ TYZ [J\]^Y_Q `Ea P Z bcd_efgQ a P a uv`wk oldxUy\ \hi bjkl` Tmn,T o n apek aR Q T ^{ef` T lim R b f (x) dx olkl| ` \}i bjkl` S e,~k u [>kVQg`ol^~k_ef` U~[ ^~ Q_^cQ}\ n Wj[J \ ^cQg`a α [J \ Q_`r[ bpd_efQga b→∞ aW \}` u\cbc`ox\ [JWYb o e5;[>k Q PZ P = Z TTmn,T nO P T [a, +∞) Z

+∞

f (x) dx = lim

b→∞

a

b

Z

f (x) dx.

a

EE~,*lwYly>gz,_ ¡v_Vf¢h£EE¡lE¤yY¥¦>£§X*¨~l§Y©y>

«¬~wlVf¬}v®§O©V¯ Z

Z

a

f (x) dx = lim

+∞

−∞

b→−∞

−∞

f (x) dx =

Z

&X£xlExlc¥ °lr±g² v Y¥¦>£ExE¡lx¶ g·¸

a

f (x) dx +

−∞

±∞,

Z

a

Z

+∞

(−∞, a]

f (x) dx.

b

a

f (x) dx, a ∈ R.

°Y³ E¢_f°~,g´<_ ¡v_Vf¢h~µ E¡lE¤>?

@AB

ª

!"

=

#$%&'()*+,)*-./

! 01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

LAPLACE

!"#&>,?@A $%&'BC( ) * D +,-./01.2(13456789: 1 -;<=>?@AB + 49C> D3E7F#>C-;<=GHID +J1.K2 1(F:2(LMNOPQ
(i)

+∞

Z

0

dx TQUV/0WX@S1;DY1:N3ZE 85<=W,GH[\1:-M<=> x2 + 1 Z

+∞ 0

dx = lim b→∞ x2 + 1

b

Z

0

dx x2 + 1

= lim tan−1 x|b0 b→∞

= lim tan−1 b = b→∞

(ii)

0

Z

ex dx.

π . 2

]^6ZDO+#E

−∞

Z

0 x

e dx =

−∞

= =

(iii)

Z

1

+∞

lim

b→−∞

lim

b→−∞

0

Z

ex dx

b ex |0b

lim (1 − eb ) = 1.

b→−∞

1 ]_`6 DI+#E dx. x Z

1

+∞

dx = x

lim

b→+∞

Z

1

b

dx x

= lim log x|b1 b→∞

= lim log b = +∞ b→∞

F#+#E\a:NI+B@A1,1.2K1(F:2(bMNOcQAdE T

!"#$% ! &'()*+,-./0 & ,1234 & 5 .67829:;! , ";<=.6> : 29?* ,-.@&45

D

!

ABC

LAPLASE

"#$%&'()*+,-E . FGH=/ I *6JK0 F 12# ( L$ M3*+1N%F &'()*OH=P

Laplace

= 6> ?@A; B : DC >3EB ;FG : >HIJK < Z[ BZG R \5UVWX?@;ABZG]B ^ 4I ;LC : _ : C

"45$ &'6( 7* ,-. D ! 89QR S : ;TU ;LMC K f :NO >PQ RST UVWX?@;YBZGM>6?

VW f : [0, +∞) → R Laplace VW L(f (t)) Z b Z +∞ L(f (t)) = F (s) = e−st f (t) dt = lim e−st f (t) dt. b→+∞

0

0

` abcDdFeJfga aZhijkl5mnaZopqrsEt F d@uf vwxPj6q@a l5yo vwaJz{a|qL}~v s ∈ R 3j xbq@x/a0cDaPfxbqLa %dFvijPdFEm3FvraaJz0a0Zz{yhr}~mnxJcDZhwd@j 1 F01 [ X 1-$P1;Y0JGZ )*+N x JcDaJz{aJ]fl5aZym3d q@amndYq@xZl5Msym3x0qij l5mn Laplace mndhi\j ] D~vx0c0z0~v l5EvrxZhqwyl5d}~v{ cZ#q@aEvaZhJjl5m3FaZEm3d (i) f (t) = 1, t ∈ [0, +∞) Z

0

PxPjMZhwx

+∞ −st

e

Z

b

e−st dt b→∞ 0 µ ¶ b 1 −st   = lim − e  b→∞ 0 s ¶ µ 1 −sb 1 = lim − e + b→∞ s s 1 = , s>0 s

1 dt = lim

1 L{1} = , s > 0. s

~cD4q@aEvaZhijkl5mnFaZEm3d

(ii) f (t) = t, t ∈ [0, +∞) Z +∞ Z b −st e t dt = lim e−st tdt b→∞ 0 0 µ ¶ b st + 1 −st   = lim − 2 e  b→∞ 0 s

"#$%&'()*+(),-. !/0 .123$#45(#"6 #)7)8 2 9 ( 45:, 6;21.+6 !

LAPLACE

µ ¶ 1 sb + 1 −sb = lim − 2 e + 2 b→∞ s s 1 = 2, s > 0 s

!"!#$%&'("

1 , s > 0. s2 < )*+$,-./#

L{t} = (iii) f (t) = sin t, t ∈ [0, +∞). Z

+∞ −st

e

0

sin t dt = lim

b→∞

L{sin t} = (v) f (t) = cos t, t ∈ [0, +∞).

0

e−st sin t dt

0

¶ b µ s sin t + cos t  = lim −e−st  b→∞ 0 s2 + 1 µ ¶ 1 sb s sin b + cos b = lim −e b→∞ s2 + 1 s2 + 1 1 , s>0 = 2 s +1

!"!#$%&'("

Z

b

Z

< )*+$,-./#

+∞ −st

e

1 , s > 0. s2 + 1

cos t dt = lim

b→∞

Z

b

e−st cos t dt

0

¶ b µ s sin t − cos t  = lim e−st  b→∞ 0 s2 + 1 ¶ µ s sin b − cos b s − esb = lim 2 b→∞ s + 1 s2 + 1 s = 2 s>0 s +1

!"#$%&'()*'(+,-.'/0 % 1

!

23+&'(45- 5)67 2 4*'(8-9: 2 %01

;<;

LAPLACE

!"!#$%&'(" s , s > 0. s2 + 1 )*+,-./0#

L{cos t} = =

(vi) f (t) = eat , t ∈ [0, +∞). Z +∞ Z b −st at e e dt = lim eat−st dt b→∞

0

= lim

b→∞

= lim

b→∞

0

Z

b

e(a−s)t dt

0

µ

1 (a−s)t e a−s

µ

L{eat } =

1?@

0

1 (a−s)b 1 e = lim − b→∞ a − s a−s 1 1 = a<s = − a−s s−a

!"!#$%&'("

>

¶ b  

1 , a < s. s−a

2A3 B 456CDEFCG D 7 H 8IDJK6L 9MHN.D :;" < #&= F > 9 :63 D ? 4 <@93K6O

APHQRBEC F=9>:

¶

Laplace

HW V IJKLIJMK NH O P&I=O.HQR IST G GXY G f, g : [0, +∞) → R LaU&O&R OVWXHQY H R O G Z Y [ L(g(t)) = G(s), s > s2 , place L(f (t)) = F (s), s > s1 Z T\H5K ^] _Q` Y KaRbU&O&R Z IJKMcH O GXY PdI=O.QH R G ISTXe Laplace H0P$e af (t)±bg(t), a, b ∈ R U&O&R*R fgKaR GXY >

1?D @ 1ES F TU

L(af (t) ± bg(t)) = aF (s) ± bG(s), s > max{s1 , s2 }.

: C!:5B,HJ\ h@9i6 ; : Dj: 1 ]kl " mXndoXpVoXqrst)*o&mgu=v / wMolux/NwM"&)*+$ygu="jw7z# )*ux{ ux/a'7^# _ Laplace d|}~)*mg~("&'\wb.)*/5 ~ +$'(yg)=#u=ojndo!#|}~\wM"&w0"nd'bo&y7q3o&-.u=/Q~("n&"&'("&(/Mszq3ux"jw0"!"!# w0 o x/a|}'(ygux"* [

!"

! 01

#$%&'()*+,)*-./

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

(i) f (t) = sinh at = 12 (eat − e−at ), t ∈ [0, +∞)

LAPLACE

= !"#$%&> ' ?()"*+,-./0123456789)3:;@ ) < A =

>?#$0@8AB ) B C C % DE'F2G4HIJK&LM)+!3N 4 %

1 L(f (t)) = (L(eat ) − L(e−at )). 2 OP#$%9,QRFS

L{eat } =

1 s−a ,

s > a,

K&LM)+!34N%

µ

1 L(sinh at) = 2

1 1 − s−a s+a

¶

./I/T , U+'FI

L(sinh at) =

a , s > |a|. s2 − a 2

(ii) f (t) = cosh at = 12 (eat + e−at ), t ∈ [0, +∞)

= !"#$%&D ' V ? )"*$,W./0XYZ!G:F2G[ 4 678A( ) E

:F@ ) \ A =]>?#$0@8A)BB C %CDE'F2G4HIJKCLT)+!G4H%

1 L(f (t)) = (L(eat ) + L(e−at )). 2 OP#$%9,QRFS

L{eat } =

1 s−a ,

s > a,

K&LM)+!34N%

1 L(cosh at) = 2

µ

1 1 + s−a s+a

¶

./I/,TU+'FI

L(cosh at) = ^FGHI_J ` N K abc

L dMe N df

OP g

hQR

L(f (t)) = F (s), s > a. hAmTx v+l+ono y3k
s , s > |a|. s2 − a 2

f : [0, +∞) → R U q;h\k

r

s t&u

T

k
ijk_iPkAh\l w

iPkAh\l

gST gST

m$inl3h&o m$inl3h&o

L(eht f (t)) = F (s − h), s > a + h.

g

g

ipq ipq"x

Laplace Laplace

!"#$%&'()*'(+,-.'/0 % 1

>

0,

!

23+&'(45- 5)67 2 4*'(8-9: 2 %01

;<=

LAPLACE

!"?@5A,#B $%&'() * +,./0/1 f (t) = eat , t ∈ [0, +∞) F (s) = L(1) = 1s , s > 23456789:;< L{eat } = L{eat · 1} = F (s − a) =

1 , s > a, s−a

=>?4 2@=>ABCD678BEFGHAIJHA9-I2@4?678F:KC < #D LH<MNO:KY ZEBFG[H" KI ()

*K\ L *KLMN O

J

^
_'`[a _ `b^Tc

dD_OcF^@e _fg

]R Q ] f : [0, +∞) → R Laplace S ghi ^ ` k l@m `3eOn7c7e _'`M^Tc 7d _KcF^@e _fgpq j R o ]QR ] Laplace L(f (t)) = F (s), s > s1 . ^bd5q n7c7e>e r9`3e f (at), a > 0 ]QR ]

1 ³s´ L(f (αt)) = F , s > as1 . a α

>

!"?@5A,B#$%&'() * sT tu=5Jbv

stw4?678F:KE < #D LH<Mx N VQJp1 T

f (t) = cos at, t ∈ [0, +∞). U LHA9=Ky1 :K6#CD6/1sz{| - JM07}PJM~ 6 JH 0.

:K 0 2@456789:;<

s s 1 ³s´ 1 a = 2 L(cos at) = F = , s > 0. s 2 a α a ( a ) + 1 s + a2

<W07-H#6p> 6 JXLHV#CD6 :>CD67LH679:K < -H0 D#LH679:K<WJM6 :K4?AF:K0#JQ1=>:;V Laplace JXAF} Tstu456789:;<WD#LH<M N VQJp1 f (t) = sin at, t ∈ [0, +∞). F (s) = L(sin t) = / . / 0 5 1 7 H L 0 # 0 D C ~ V b J ~ 6 H J i < # @ < p 8 M J / 0 s N 6 3 < w z H L 9 A K : 0 @ 2 5 4 7 6 9 8 :;< 1 s2 +1 ,

s>0

a 1 1 ³s´ 1 = 2 = L(sin at) = F , s > 0. s 2 a α a ( a ) + 1 s + a2

ZEBFG[H" IK()

J

*K\ L *XY J NO

]

^
_'`[_a`b^Tc

dD_OcF^@e _fg

]R Q ] f : [0, +∞) → R Laplace S ghi ^ ` k l@m `3eOn7c7e _'`M^Tc 7d _KcF^@e _fgpq j R o ]QR ] L(f (t)) = F (s), s > s1 . Laplace ^bd5q n n7c7eOe rF`3e t f (t) ]QR L(tn f (t)) = (−1)n F (n) (s), s > s1 .

!"

#$%&'()*+,)*-./

! 01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

LAPLACE

!"#&>,?@A $%&'BC( ) =

D*+,-./01

D2+,3.456789:E; F <=> : ?@AB/CD

(i) f (t) = t cos at, t ∈ [0, +∞). F (s) = G <EFG9 - HD 894(IJ4#DHK,LMN/ O5PQ/>R 4 /=S: T(:06B/NO#?*4UVWX: K,<EF789O s L(cos at) = s2 +a2 , s > 0.

Y0Z3 456G89:

L(t cos at) = (−1)1

µ

s s2 + a2

D*+[-./N1

¶′

=

s2 − a2 , s > 0. (s2 + a2 )2

D*+[3.4567\ 8 :H;F <E:N?9AB/CD

f (t) = teat , t ∈ [0, +∞). F (s) = L(eat ) = (ii) G <=FG-9DH8\4(I54#DHK,LM/NO5] P />^ 4 /=:ST;:X6B/NO#?*4_V\:XK,<EFG89O`Y03Z456G89: 1 s−a , s > a. at

L(te ) = (−1) (iii)

1

a@LEb(T;4CcdI O ;F
µ

1 s−a

¶′

L(t2 e−αt ) = −

=

1 , s > a. (s − a)2

2 . (s − α)3

f ?KLgM" 9N h J & O )Pi Q ) RS T U kW V mnoWp r=sp t xMy0o\kNz z{|}m l m f : [0, +∞) → R l j q uvjiw j ~ X M o r kSm |}m>k z5|9G k0p |C r= xXm>k5pW l Laplace L(f (t)) = F (s) q q T q jZ Y j kNz z jZ j Y 1 L(f (t)) = 1 − e−sT =

!"[\&>,?($}&h )

D*+[-./N1

f (t) =

½

Z

T

e−st f (t) dt, s > 0.

0

AB/E:F 1 0 ≤ t ≤ 12 e#O#D f (t) = f (t + 1), t > 0. 1 0 2
-67LEb5<M/=FG- F :>?LEO#DWI5:0O^IJO5<EO(IJb5LM1 1 L(f ) = 1 − e−s

Z

0

1

e−st f (t) dt

!"#$%&'()*'(+,-.'/0 % 1

!

23+&'(45- 5)67 2 4*'(8-9: 2 %01

1 = 1 − e−s =

;<=

LAPLACE

1 2

Z

e−st dt

0 −st ¯ 12

e ¯ 1 ¯ 1 − e−s −s 0 1

1 − e− 2 s = . s(1 − e−s )

!" #$%&

' ('

)*

+,-.+,/- 0* 1

23+451 *67 +,8 LaplaLM1A
f : [0, +∞) → R )MIJN 1J7 7 ) : = < 0 > ? A 1 = < 1 B C OLDCEF 7G+,H 2 @ ) ; )(K ce L(f (t)) = F (s), s > s1 9 k ; N : B R F L . L A 1 = < 3 I 6 6 * U T / C < C = < 1 * f (m) , m = 0, 1, 2, . . . , k ;SK O ; ) ; +∞ V P -P7=IJ1J7 ; +,-/*M1 )MN 2J+#15*A7 ) +Y8Z? Laplace */2E? f (k)(t) IJ1J7=7 )MN [\-P7 N >?@A

BC

D

EF

EGHI J

KL

L(f (k) (t)) = −f (k−1) (0)−sf (k−2) (0)−. . .−sk−1 f (0)+sk F (s), s > s1 .

]^_3`abcd5ef_3ghiJdZj=iJk\lNmAecdan=opqAn=drqAoEiJs5l#m

QR

L(f ′ (t)) = −f (0) + sF (s)

tuvuw

L(f (2) (t)) = −f ′ (0) − sf (0) + s2 F (s).

&D"u& xy$z& {& ' Q|}npb6~d y(t) mW_3v{{dNmAk\m/w bc d Zw vJ#iJgBwtumAZQn~ n=d y′ + 2y = e−3t tuvuw (i) wtuvJehi{_3iuwGm/ vJghowQtu n=s\ePNBtud y(0) = 1. v {ghmNm/%il#mWb/vJn=oEd5l#v{bBwn=l#`J bcd5 y. Laplace Q|}npb6~ L(y(t))) = F (t). vughehi5eSb6vJb6il#mWb6vJn=opd\lNv{bZwn=lN` Laplace qAoEiJs5l#m 5T,?UV

S

6W

X

Y

X

Z[

\

|y_3m/wGh

L(y ′ + 2y) = L(e−3t ) ⇒ L(y ′ ) + 2L(y) = L(e−3t ). L(y ′ ) = −y(0) + sF (s)

−y(0) + sF (s) + 2F (s) =

tuvuw

L(e−3t ) =

1 s+3 ,

s > −3,

qAoEiJs5l#m

1 1 ⇒ (s + 2)F (s) = +1 s+3 s+3

!"

#$%&'()*+,)*-./

! 01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

LAPLACE

s+4 s+3 s+4 . ⇒ F (s) = (s + 2)(s + 3) ⇒ (s + 2)F (s) =

(ii)

= !"#$%&' (

y(t)

)*+,-./.(012)345)6789 % : ( ;<= 7 ->?@A>BC7DEFG:)3H
EF-F727 EF->NOA.+,AF7J)6I@%CP7 QR->BOSM7DEFT3U Q #MV5NW12GCEF)WQ A[\@)*%&->#MS$(5\2-.%<]7 #M\2^>Q

= !"#$%&'

Laplace

\@)

L(y(t))) = F (s).

%O(5Q

y(0) = 0

EF-F7

y ′′ + 2y ′ − 3y = 1 X > -?@YBO)Z18)6I y ′ (0) = 1

y.

A_-FI`BONOAaN0%6->b Q %&Ac\@)*%&->#MS$(5\2-.%<7]#M\2^

Laplace

T3d S A>V

BC

L(y ′′ (t)) + 2L((y ′ (t)) − 3L(y(t)) = L(1) ⇒ −y ′ (0) − sy(0) + s2 F (s) + 1 s 1+s ⇒ (s2 + 2s − 3)F (s) = s 1+s ⇒ F (s) = s(s2 + 2s − 3) 1+s ⇒ F (s) = . s(s − 1)(s + 3)

2(−y(0) + sF (s)) − 3F (s) =

D Ee D

%$FGHIJKghLF I , i M&NOj M k:P l QB. I mjh RS@n l mTIKopq h lnrjk f

stiMopu h v l rjk

D Ee D

UV W

XY

ewx

z y

z

L(f (t)) = F (s) ]dM, u F (s). 9 y

Z

F (s), s > s1 {

9d6z}>>}>
y

z

| }

3@z6 y~

Laplace

u6z}

[\ y

y

>@}az3 y

f (t) = L−1 (F (s)). = bB9-

L(f (t))) = F (s) ⇔ L−1 (F (s)) = f (t).

z&6z

}

f (t) < Laplace

z&$

!"#$%&'()$%*+,-./0, 1 ( 234-$%-( !+5!6 2 #$%&7(028,0(

9:;

LAPLACE

& ? @ A' @ '() B*+:C,-.= " D> ?/>E/F 0G>HIJ1 K $+LD>=M &)2> EI+NOIJK34PFQ#> $5&"R # ?67 4 S $ 8 > 9- : & 4PIJT < !"=#> $%) U V < ; =>?@A EF;GHIJKW= L X MNH OPH = =YXQR M = TU;>V= J ?XT6JY=ZH T [ Laplace BCD B D BPS WZ[ W B Laplace J \ O];.H^_,MP[ M]=VJXH H]JX_X` EF;bcdHeOfHe`Pg= ?,=V;bch= JXM]=Zi = j k T:;g=VJ ?lTmJY=gH T [ B B B a,BPS BPS W B BfS WZ[ W B Laplace n )m'

L−1 (aF (s) ± bG(s)) = aL−1 (F (s)) ± bL−1 (G(s)).

o 'P\RM

L−1 (F (s)) = f (t)

=g`P=V;

L−1 (F (s − a)) = eat f (t).

]pP ' \RM

L−1 (F (s)) = f (t)

=g`P=V;

1 L (F (as)) = f a −1

< $52$+C^FHM q:4r!$ >#$ ' (i)

µ ¶ t . a

_ s`a#t]uvwxu>yz{NsX|P}C~y }]tf{Xx{X 6u.}>sX,!Y6s#}/,xX

Laplace

}fY7,Y|fXtP}]F,F

s−1 . s2 + 4 b t]F6 6{#l{2~d|P}>sX}/ //}fC}]uZ}>{XsX|P}Cy~}]tf{Xx{XK6u.}gsX,Fxs#}C ,x{X La2s27}>{Y| 6u.}gsX,Fxs#}C,x }gd|3,Y|fsXtP}]F,uGd| 2s2 place Laplace cos at Z!{XY6u sin at µ µ µ ¶ ¶ ¶ s−1 s 1 −1 −1 −1 L = L −L s2 + 4 s2 + 4 s2 + 4 1 = cos 2t − sin 2t. 2 F (s) =

(ii)

_ scaztfuvw%ugyC{sX|P}Cy}ft]{X6{XL6u.}gsX,Fxs#}C,xX F (s) =

1 . s2 − 3s + 2

Laplace

}]FQ,Y|fXtP}]F,F

!"

! 01

#$%&'()*+,)*-./

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

LAPLACE

!"#$%&'()*%+',%-.()/01( 234 $ 5623$%562789$%:);<$4$%='>?@?,6ABCD()E / /F89G/ H; ='CIJK,%L - MN&'CO:)23(),?JK,92P,%Q.:R$%(#/ AST2 $UVG,OW = X- JK,X@?,6AS2P:R()Y * ,XZ@ ,?8%@?C.&[\](#>?V^_ \ ` = &[C6Vab]=W:R-.='$%&aV[cO:Rd/ \]=Z=e >7]Mf,%-.()/

,?JK*6V[? / @?&6;g:f89,%-O()/

1 1 = , s2 − 3s + 2 (s − 1)(s − 2)

A

32 $ klJ%*6sV'/

= 1

−1

L

(iii)

:RC.o

B

V[hFVi,92T$jb]:fV[/

A B 1 = + . (s − 1)(s − 2) s − 1 s − 2

89$92f>%&'$ A

89$92

h^MN,%-O()/ µ

1 = A(s − 2) + B(s − 1). 1 = −A ⇒ A = −1

1 s2 − 3s + 2

¶

= −L

1 s−1

¶

$ Z&[/mn1/i;#,Y$%=aV6;g:fV[&',%"),%op(#/qVG$%:RMfC.()$?V62P:R()*Ko

C D@

rJK/G235'c

s = 2

−1

+L

h^Mf,%-.()/ µ

1 s−2

Laplace

1 = B ¶

V'C.op:R-.='>%&aV[CEF

1 . s2 − 6s + 9

1 1 = s2 − 6s + 9 (s − 3)2

h^MN,%-O()/E*6VX2 −1

L (F (s)) = L

:RC.o

µ

= −et + e2t .

F (s) =

(iv)

−1

eBA 2T$

−1

µ

1 (s − 3)2

¶

$ ?&'/mn#/G;#,4$%=aVX;g:BV[&',%"),%s o ()/FVi$%:RMfC.(#$?VX2P:R(#*%o

C G@

F (s) =

s . (s + 1)3

= te3t . Laplace

V[COop:R-.='>%&aV[CEF

e

!"#$%&'()$%*+,-./0, 1 ( 234-$%-( !+5!6 2 #$%&7(028,0( <

9:;

LAPLACE

!"#$%&'()*+,-./01

s+1−1 s = (s + 1)3 (s + 1)3 s+1 1 = − (s + 1)3 (s + 1)3 1 1 = − . (s + 1)2 (s + 1)3

<

2,345

µ

1 L−1 (F (s)) = L−1 (s + 1)2 1 = te−t − t2 e−t . 2 (v)

¶

− L−1

= >? 5

637+89*+:;.%,5&<=/0; >$/737%&?@%&AB)@+C/:5&>DE # FG)@56/.1H>D)*-&A

F (s) =

µ

1 (s + 1)3

Laplace

¶

/4FGI A >D'G<4J&3=/7F(>DF(A

1 . s2 − 2s + 5

!KLM+:1NO7P

1 1 = s2 − 2s + 5 (s − 1)2 + 4 QR#E%&'G)@+S-./01

L−1 (F (s)) = L−1 (vi)

= @? 5

µ

1 (s − 1)2 + 4

¶

1 = et sin 2t. 2

637+89*+:;*%T5&<=/0; U > /437%&?@%&V A )@+C/W5&>D#$F()*56/01X>D)*-&A

>DF(A

C

F (s) = 3.; >$Z[%&'()*+

\

A B

Z[5[1

C

Laplace

2s − 1 . s(s − 1)(s − 2)

/7QC/W%[] 1 5^_`>$/4+

A B C 2s − 1 = + + . s(s − 1)(s − 2) s s−1 s−2

/4FGY A >D'(<7J&3=/4F

AB

"#$%&'()*+(),-. !/0 .123$#45(#"6 #)7)8 2 9 ( 45:, 6;21.+6 !

LAPLACE

!"#$%&#'()*"+ ,( -.+/0123!"435)6- 7 + 89:;%"<=>?5 <

A(s − 1)(s − 2) + Bs(s − 2) + Cs(s − 1) 2s − 1 = s(s − 1)(s − 2) s(s − 1)(s − 2)

@A*A-;!"23*

2s − 1 = A(s − 1)(s − 2) + Bs(s − 2) + Cs(s − 1) ⇔ 2s − 1 = (A + B + C)s2 + (−3A − 2B − C)s + 2A  =0  A+B+C ⇔ −3A − 2B − C = 2  2A = −1   B + C = 12 2B + C = − 12 ⇔  A = − 12   C = 32 ⇔ B = −1  A = − 12 .

!"#$ −1

L

6@BC:

µ

(vii)

2s − 1 s(s − 1)(s − 2)

¶

µ ¶ µ µ ¶ ¶ 1 1 1 −1 1 3 −1 −1 = − L −L + L 2 s s−1 2 s−2 3 1 = − − et + e2t . 2 2

! $"#%"&'()*+' ,-./$01234,5673#"&.089.0:;<='>3?$06@A7BC<=D$ 3EFG6@<=H0:

5LMA7.0IC<='

F (s) =

s+2 . s2 − 2s + 5

s+2 s+2 = s2 − 2s + 5 (s − 1)2 + 4

Laplace

3&BC:;6@IC1&J0"K3#B

%$ !"#$%&'()$%*+,-./0, 1 ( 234-$%-( !+5!6 2 #$%&7(028,0(

LAPLACE

9:;

s−1 3 + (s − 1)2 + 22 (s − 1)2 + 22 2 s−1 3 = + (s − 1)2 + 22 2 (s − 1)2 + 22 3 = et cos 2t + et sin 2t. 2 05>?893!"#$%&'$()*+,-./01)23450 "#+-67+-89:'$;0(-3<=4>?:7!01@A3<:7BC8 (viii) Laplace 05F> 3<>?8 =

F (s) = "1)234GH+-D?:'$

A

A B

!"#"$

C

%&'(%)*"+$ #,-./01% 2

1 . s(s + 1)2

A Bs + C 1 = . + s(s + 1)2 s (s + 1)2 34561*786 %9#5:3;% $<:=>?@&45A&29$BC : 'DE0*5FGHI2 A(s + 1)2 + (Bs + C)s 1 = s(s + 1)2 s(s + 1)2

!"#"$045@&#

1 = A(s + 1)2 + (Bs + C)s ⇔ 1 = (A + B)s2 + (2A + C)s + A   A+B =0 ⇔ 2A + C = 0  A =1   B = −1 ⇔ C = −2  A = 1.

JK>5LM%&2 −1

L

µ

1 s(s + 1)2

¶

µ ¶ ¶ µ 1 s+2 = L −L s (s + 1)2

!

"#$%&'()*+(),-.

! /0

"#

.123$#45(#6 )7)8 2 9 ( 45:, 6;21.+6

LAPLACE

µ ¶ µ ¶ µ ¶ 1 s+1 1 = L −L −L s (s + 1)2 (s + 1)2 = 1 − e−t − te−t .

!?@"A # BCD$%&% ' EF( G )*H % IAJKLMN'*O +% #PJ,Q # N-.KL/0RNSTUG 0 VN T-'($ 8 # %W1 B 2 0 X ' T*%+3 # 4 0 5

< =>

Laplace

6 89:;<$=->1?: @ABCD8 EFGHIJK1:?LM>NOCP*Q 7 G BRO?SH>1I&BT:?<$UVW= >NX: BY8Z<$>NO?[ IR\]@M:F8-=HI&^_\9`aSW<$= Laplace bY89:?K1O?LY8DcE de@fIJgY8Z<*de<$Ihie@aj klSCBMmnP1\]@MI&BT:F8YXZ LY\DNI&BT:?<$UV=W>N:XBC8r<$>Nm Bs=-o/tXP*@sie1I&BR:?<$U_=->1:XBY8r<$>1m j Laplace

'()F'%D+N\] z{02X ' TX' [

(i)

^

:|`aSR}NIR\"=~bY89:?KNO?LC8DEFG~IJgY\D<*ie<$=

y ′ − 2y = 1

$>1I

y(0) = 1.

K1:?LM>1m?MO-@ABT:?o;BT e O >NIBR:?<$UV=W>1:XBY8r<$>1m 1I

EF:F8Vt?LM:

L(y ′ − 2y) = L(1) ⇔ L(y ′ ) − 2L(y) = L(1) ⇔ sL(y) − y(0) − 2L(y) = L(1) ⇔ (s − 2)L(y) = y(0) + L(1) 1 ⇔ (s − 2)L(y) = 1 + s s+1 ⇔ L(y) = s(s − 2)

_

y(t) = L

LY\DNI

A

EF:F8

B

Bsv&BROF89:yde
−1

µ

¶ s+1 . s(s − 2)

A B s+1 = + , s(s − 2) s s−2

!"#$%&'()*+,-./0123*45)63789 !#*:$;6<$%9/637'=>9?@AB 0 =>9C@D,E"$ F# &FG = H , @DI* $%=J0

KLM

!"#$%&'(

s + 1 = A(s − 2) + Bs. N )*+ ,-./01234( 5 N )6+ O ,7./01234( s=0 1 = −2A ⇒ A = − 12 s=2 3 = 2B ⇒ 3 589:"0$%&;( B= 2 µ µ ¶ µ ¶ ¶ s + 1 1 1 1 3 L−1 = − L−1 + L−1 s(s − 2) 2 s 2 s−2 1 3 = − + e2t . 2 2 3G( P +<=!1>?4(@ABCDE%)*+0FG0H%)IJKLD(7M%AINOPQRN C ′ (ii) 2y − y = e2t y(0) = 1. STFG+0H;3G$0U;VWXY& +0Z[&Y\34(]&@+0NR./C23G+!&%)^NR3G$ N/&;C2W_"#+0H;+!"#`0W'P )^NR$%&'C%&Y+ Laplace ,-a . 01V3G(

L(2y ′ − y) = L(e2t ) ⇔ 2L(y ′ ) − L(y) = L(e2t ) ⇔ 2sL(y) − 2y(0) − L(y) = L(e2t ) ⇔ (2s − 1)L(y) = 2y(0) + L(e2t ) 1 ⇔ (2s − 1)L(y) = 2 + s−2 2s − 3 ⇔ L(y) = (2s − 1)(s − 2) JK+K)/`0H;+

y(t) = L

Q

H%AIN/JK01234(

A

JK+K)

B

−1

µ

&'b, &@K)*c + dTN/&;(

¶ 2s − 3 . (2s − 1)(s − 2)

2s − 3 A B = + (2s − 1)(s − 2) 2s − 1 s − 2 JK+K)/`0H;+

2s − 3 = A(s − 2) + B(2s − 1).

!"

=

! 01

#$%&'()*+,)*-./

!"

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

>=

#$%&'()*+,-

.

/"

LAPLACE

#0%1'(2 ) +3-

s = 12 −2 = − 32 A ⇒ A = 43 s=2 1 = 3B ⇒ 1 .4567(89:;B= 3 µ µ µ ¶ ¶ ¶ 2s − 3 1 1 4 1 L−1 = L−1 + L−1 (2s − 1)(s − 2) 3 2s − 1 3 s−2 µ ¶ 1 −1 1 1 4 L ( ) + = L−1 6 3 s−2 s − 12 = ?

2 1 t 1 2t e2 + e . 3 3

"<=>)?@AB - CDE
(iii) y ′ (0) = 1.

y ′′ − 5y ′ + 6y = et ,

SQG,"(H;+,8(T;'()2+3-U:?'V+,W - :?"(OR%&E*+,"X:9YOR+38 #$%&'()*+,-

Laplace

+3-

y(0) = 1

O1:;E2Z[7\"(] H "X7\^(Z]P

JK"K

_OR89:]E9:B"

L(y ′′ − 5y ′ + 6y) = L(et ) ⇔ L(y ′′ ) − 5L(y ′ ) + 6L(y) = L(et ) ⇔ s2 L(y) − sy(0) − y ′ (0) − 5(sL(y) − y(0)) + 6L(y) = L(et ) ⇔ (s2 − 5s + 6)L(y) = sy(0) + y ′ (0) − 5y(0) + L(et ) ⇔ (s2 − 5s + 6)L(y) = s + 1 − 5 + ⇔ L(y) =

s2 − 5s + 5 (s − 1)(s2 − 5s + 6)

s2 − 5s + 5 . ⇔ L(y) = (s − 1)(s − 2)(s − 3) JK"K1^(H;"

y(t) = L

−1

µ

s2 − 5s + 5 (s − 1)(s − 2)(s − 3)

¶

1 s−1

!"#$%&'()*+,-./0123*45)63789 !#*:$;6<$%9/637'=>9?@AB 0 =>9C@D,E"$ F# &FG = H , @DI* $%=J0 N

!"#$%&'()*+,-.

/

A B

&'0'1

C

234526('17089:$%;2 .

A B C s2 − 5s + 5 = + + . (s − 1)(s − 2)(s − 3) s − 1 s − 2 s − 3

O<=)>3(?>@260)A
s2 − 5s + 5 = A(s − 2)(s − 3) + B(s − 1)(s − 3) + C(s − 1)(s − 2) ⇔ s2 − 5s + 5 = (A + B + C)s2 + (−5A − 4B − 3C)s + 6A + 3B + 2C  =1  A+B+C ⇔ −5A − 4B − 3C = −5  6A + 3B + 2C = +5  =1−A−B C ⇔ 2A + B = 2  4A + B = 3   C =1−A−B ⇔ B = 2 − 2A  2A = 1   C = − 12 B =1 ⇔  A = 12 ,

KLM

!"

#$%&'()*+,)*-./

!01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

LAPLACE

!"#$%&'(

¶ 2 s − 5s + 5 L−1 (s − 1)(s − 2)(s − 3) µ µ µ ¶ ¶ ¶ 1 1 1 1 −1 1 −1 −1 +L − L = L 2 s−1 s−2 2 s−3 1 1 = et + e2t − e3t . 2 2 µ

(iv)

=

)*+,-./01( 23&.4567859:& ;8<=> ) &1?@ABC%DE)FG=FH%DIJ#K@ALM( NOPD 53K@56(Q?@A

y′ + x = t x′ − y = 1

<=(

JR)RD y(0) = 1 x(0) = 1. S@G=)FH:<=$FT:UAV&1)FWX&1Y<=Z( &1)F56[9;8\ < )!&ODP56<\$ 59&%D]^ W "#)FH:)!"#_FAV? DP56$%&';%&'(MW Laplace M` a [ F-U<=(^D]56FC:7UA')F<\)

)

L(y ′ ) + L(x) = L(t)

⇔

)

sL(y) − y(0) + L(x) = L(t)

⇔

)

⇔

)

sL(y) + L(x) =

1 s2

sL(x) − L(y) =

1 s

⇔

)

L(x) =

1 s2

L(y) =

s−1 s2 +1

L(x′ ) − L(y) = L(1)

sL(x) − x(0) − L(y) = L(1) sL(y) + L(x) = L(t) + 1 sL(x) − L(y) = L(1) + 1

+

s+1 s2 +1

+1 +1

!"#$%&'()*+,-./0123*45)63789 !#*:$;6<$%9/637'=>9?@AB 0 =>9C@D,E"$ F# &FG = H , @DI* $%=J0

(v)

M

⇔

)

⇔

)

⇔

)

KL

x = L−1 ( s12 ) + L−1 ( ss+1 2 +1 ) y = L−1 ( ss−1 2 +1 ) x = t + L−1 ( s2s+1 ) + L−1 ( s21+1 ) y = L−1 ( s2s+1 ) − L−1 ( s21+1 ) x = t + cos t + sin t y = cos t − sin t.

!"#$%&'(%)*+%,-./01.*2+ 3456!+789:;<=>?@AB,@C=>DEFGHI: (JK=>LM. G9./(N89: y ′′ − 4y + x = e−t

5B(

x′′ − x + y = e2t

OE O> x′(0) = −1, y′(0) = 2 P Q9AB@CR5BS@TR,1:U+7@VW+7,X5BY( +7@./Z*3456#+[\> ./56S Laplace .*+=>LV]^F@CR#^F_@:U8 >\./S=+23=+2(JV `JaZ ,@$15B(]>L./,@
)

L(y ′′ ) − 4L(y) + L(x) = L(e−t ) L(x′′ ) − L(x) + L(y)

= L(e2t )

s2 L(y) − sy(0) − y ′ (0) − 4L(y) + L(x) = L(e−t )

⇔

)

⇔

)

(s2 − 4)L(y) − s − 2 + L(x) =

1 s+1

(s2 − 1)L(x) − s + 1 + L(y) =

1 s−2

⇔

)

(s2 − 4)L(y) + L(x) =

1 s+1

+s+2

(s2 − 1)L(x) + L(y) =

1 s−2

+s−1

s2 L(x) − sx(0) − x′ (0) − L(x) + L(y)

= L(e2t )

!"

#$%&'()*+,)*-./

!01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

LAPLACE

 1  [(s2 − 1)(s2 − 4) − 1]L(y) = (s2 − 1)[ s+1 + s + 2]    1 − s−2 −s+1 ⇔    1  (s2 − 1)L(x) = s−2 + s − 1 − L(y) [(s2 − 1)(s2 − 4) − 1]L(y) = (s2 − 1)(s + 2) −

⇔

)

⇔

  [(s2 − 1)(s2 − 4) − 1]L(y) =

(s2 − 1)L(x)

 (s2 − 1)L(x)

=

=

=

1 s−2

(s2 − 1)L(x) =

1 s−2

⇔

)

⇔

)

L(y) =

1 s−2

L(x) =

1 s+1

⇔

)

1 y = L−1 ( s−2 )

⇔

)

L(y)

1 x = L−1 ( s+1 )

y = e2t x = et

1 s−2

1 s−2

+ s − 1 − L(y)

(s2 −1)(s2 −4)−1 s−2 1 s−2

+ s − 1 − L(y)

+ s − 1 − L(y)

!"#$%&'()*+,-./0123*45)63789 !#*:$;6<$%9/637'=>9?@AB 0 =>9C@D,E"$ F# &FG = H , @DI* $%=J0

f (t)

L(f (t)) = F (s)

1 s

s>0

eat

1 s−a

sin at

a s2 + a 2

s>0

cos at

s s2 + a 2

s>0

ebt sin at

a (s − b)2 + a2

s > |b|

ebt cos at

s−b (s − b)2 + a2

s > |b|

tn tn eat

n! sn+1

s>a

s>0

n! (s − a)n+1

f (at)

1 s F( ) a a

eat f (t)

F (s − a)

s>a

KLM

"#$%&'()*+(),-. !/0 .123$#45(#"6 #)7)8 2 9 ( 45:, 6;21.+6 !

f (t)

L(f (t)) = F (s)

f ′ (t)

sF (s) − f (0)

f ′′ (t)

−f ′ (0) − sf (0) + s2 F (s)

LAPLACE

f (k) (t) −f (k−1) (0) − sf (k−2) (0) − . . . − sk−1 f (0) + sk F (s) tn f (t) t sin at t cos at

(−1)n F (n) (s) 2as (s2 + a2 )2 s2 − a 2 (s2 + a2 )2

s>0 s>0

!"#$%&'()*+,-./0123*45)63789 !#*:$;6<$%9/637'=>9?@AB 0 =>9C@D,E"$ F# &FG = H , @DI* $%=J0

L−1 (F (s)) = f (t)

F (s)

1 s

s>0

1 s−a

s>a

eat

a s2 + a 2

s>0

sin at

s s2 + a 2

s>0

cos at

a (s − b)2 + a2

s > |b|

ebt sin at

s−b (s − b)2 + a2

s > |b|

ebt cos at

n! sn+1

s>0

n! (s − a)n+1

s>a

tn tn eat

1 s F( ) a a

f (at)

F (s − a)

eat f (t)

KLM

!"

#$%&'()*+,)*-./

!01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

F (s)

L−1 (F (s)) = f (t)

sF (s) − f (0)

f ′ (t)

−f ′ (0) − sf (0) + s2 F (s)

f ′′ (t)

−f (k−1) (0) − sf (k−2) (0)

f (k) (t)

− . . . − sk−1 f (0) + sk F (s) (−1)n F (n) (s)

tn f (t)

2as (s2 + a2 )2

s>0

t sin at

s2 − a2 (s2 + a2 )2

s>0

t cos at

LAPLACE

!"#$!%&'()*+,-()./01(

2$2

!"#6$7 %&8# 9'()*

3 45

+ ',-.)/01. 23456789:. ;<=>?@.:ABC&DEF; 6G6HI'6J?KIJLM,-NOCP+

: ;<

Z

Q<41+K; R S =

f (x) =

1 x2

f (x) =

1 x3

+∞

f (x) dx 1

f (x) = e−x

3

f (x) = xe−x >

f (x) = x2 e−x

+ J,F.)/P.5T2 U6 C&V. 41+KW9XYZMCP+J4H=[W9CPQK\

: ?< 5

f (x) = 1 + 2x @

f (x) = 2 − x2

A

f (x) = 2 sin x − cos 2x + 1 B

f (x) = x2 − x + 2

Rb

f (x) = xe2x

Laplace

41NO;]^K+K,F+J?@_J41` N W9AB;F+K,a4FLMW9.:NO;

!"

#$%&'()*+,)*-./

! " =>

?@

,A

!01

"#

/234%$56)$7 *8*9 3 : ) 56;- 7<32/,7

f (x) = e−2x sin 5x f (x) = xe3x cos x f (t) =

n

f (t) =

n

f (t) =

n

#$%$& a 0≤t≤1 f (t) = f (t + 2), t > 0. ,a∈R 0 1 0. −1 1 < t < 2 t 0 ≤ t ≤ π #$%$& f (t) = f (t + 2π), t > 0. 2π − x π < t < 2π

' D()*+,-.+/0123'$456780 9:;6 )*2$<.2$=>?.@+ 6/'$9!A:BC?D'(6EFG9!?DHI= B C

9!RC4*'$)56;ST9!U+ KV4 =E

=G

=H

"

":W

LAPLACE

F (s) =

1 s+6

F (s) =

1 s2 + 5

F (s) =

1 +3

2s2

F (s) =

s s2 + 2

F (s) =

1 s2 + s + 1

F (s) =

s+2 s2 + s + 1

Laplace

6JKL4MN$'$)*'(OPQ(6JK

!"#$!%&'()*+,-()./01(

! 5

6

234

F (s) =

1 s2 + 6s + 11

F (s) =

1 s2 + 5s + 6

F (s) =

s−3 s2 + 5s + 6

" $%&'()*+,-./*01234"+56"78097:;< =>?1 "+@A*+5B1C80D;EFGHIB1JK2LMK!G'N1 EFOAP # G :QR879 *+S'()G'1TU " :V*+&#OA:G ; :V"+K!WXR-Y O "7:>1JK!OA*4, 7

Laplace.

<

y′ + y = x

=

y(0) = −2

Z]\

y ′ + y = cos x > ?

y ′ + 3y = ex y ′ − 2y = 1

@

Z]\ Z]\

y ′ − 2y = e−x

^ 5

Z[\

5_` 5

y ′′ + 4y = 0

Z[\

y(0) = 3

y(0) = −1 y(0) = 1

Z]\

y(0) = 1

y(0) = −1, y ′ (0) = 1

y ′′ − 6y + 25 = 0 y ′′ − 2y ′ + 5y = 0

5A5 5B6 5C<

Z]\ Z]\

y(0) = −1, y ′ (0) = 1 y(0) = −1, y ′ (0) = 1

y ′′ − 3y ′′ + 3y − y = 0 y ′′ + 5y ′ + 6y = 0 2y ′′ + y ′ − 3y = 0

Z]\ Z]\

Z]\

y(0) = 1, y ′ (0) = 1, y ′′ (0) = 1

y(0) = 0, y ′ (0) = −1 y(0) = 1, y ′ (0) = 1

!"

#$%&'()*+,)*-./

=>? =@A

y ′′ + 4y = 0

=>B

#"

y ′′ + y = x2 + x

$ %&

#"

y ′′ + 2y ′ + y = 2ex

'

!"

y(0) = 0, y ′ (0) = 1

y(0) = 0, y ′ (0) = −1

!"

D>D

y ′′ − 4y ′ + 3y = 3e5x

!"

DE?

y ′′ − 2y ′ + 5y = cos x

#"

!"

y(0) = 1, y ′ (0) = 0

!"

y(0) = 0, y ′ (0) = 0, y ′′ (0) = 1

y(0) = 0, y ′ (0) = 0

y ′′ + 2y ′ + 5y = cos 2x + 2 sin 2x

D*A

y ′′ − 2y ′ + 5y = x − cos x + ex

DEB DEC

y(0) = 4, y ′ (0) = −3

y(0) = 1, y ′ (0) = 1

y ′′′ − 3y ′ + 3y ′ − y = 2ex

D@F

!"

y(0) = 2, y ′ (0) = −1

y ′′ + 2y ′ + y = (x2 − 1)ex

DE=

LAPLACE

y(0) = −2, y ′ (0) = 2

y ′′ − 2y ′ + y = x3 − 2x2 + x − 2

D

D

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y( π2 ) = 0, y ′ ( π2 ) = 1

y ′′ − 4y ′ + 5y = 2x

=>C

D

!"

y ′′ + y ′ + y = 0

! 01

#"

!"

y(0) = 0, y ′ (0) = 0

y(0) = 0, y ′ (0) = 0

y ′′ − 5y ′ + 6y = ex − 3 cos 2x + 2x3 − 1

!"

y(0) = 0, y ′ (0) = 1

y ′′′ − y ′′ + y ′ − 1 = ex + 2 cos x + x − 1

!"

y ′ (0) = 1, y ′′ (0) = −2

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y ′ + 2x = e−t x′ − 2y = et

!"#

x(0) = 0 $ y(0) = 0.

5%&

y ′ + x′ = t

!"#

x′′ − y = e−t x(0) = 3 $ y(0) = 0 $ x′ (0) = −2.

' 5

2y ′ + x′ − 3x = −e−2t

!"#

2x′ − 4y − 3x = 3e−t − 3e−2t x(0) = 3 $ y(0) = 0.

5)6

y′ + x = t

!"#

x′ − y = 0 x(0) = 0 $ y(0) = 1.

578

y′ − x = 0

!"#

y − x′ = 0 x(0) = 1 $ y(0) = 1.

595

x′ + y = 0

!"#

x − y ′ = −2t x(0) = 0 $ y(0) = 1 (

!"

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! 01

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2x − 3y ′ = 0

!"

x′ − 2y = t x(0) = 1 # y(0) = 0 $

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x′ − y = 0 x + y′ + z = 1

!"

x − y + z ′ = 2 sin t x(0) = 0 # y(0) = 1 z(0) = 0.

"#

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LAPLACE

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http://www-ee.stanford.edu/ http://www.registrar.fas.harvard.edu/Courses/Engineering Sciences.html

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