home
***
CD-ROM
|
disk
|
FTP
|
other
***
search
/
Multimedia Differential Equations
/
Multimedia Differential Equations.ISO
/
diff
/
chapter8.1p
< prev
next >
Wrap
Text File
|
1996-08-13
|
11KB
|
491 lines
à 8.1èDefïition ç ê LaPlace Transform
äè Fïd ê LaPlace Transform ç ê given function
â èèFor f(t) = tì,èïtegration by parts (twice) gives
èèèèèè ░▄èèèèèèèèèèètìèè │bèè2è░▄
è ÿ{ tì } = ▒ètì eúÖ▐ dtè=èlim - ── eúÖ▐│è+ ──è▒ t eúÖ▐ dt =
èèèèèè ▓╙èèèèèèèèb¥∞èèsèè │╠èèsè▓╠
è 2èèèètèè ▒bèè2 ░▄èèèèèè 2èèèè1èèè▒bèè2
è── lim - ── eúÖ▐▒è+ ── ▒èeúÖ▐ dt =è── lim - ── eúÖ▐ ▒è= ───
è s b¥∞èèsèè ▒╠è sì ▓╠èèèèèèsì b¥∞èèsèèè▒╠èèsÄ
éS The LAPLACE TRANSFORM is used ï many areas ç applied
maêmatics ë convert an unsolvable problem ë an equiva-
lent solvable problem.èThis is a three step process
1) Transform ê problem as given ï terms ç a function f
with variable t, ë a function F given ï
terms ç a variable s i.e. go from f(t) ë F(s)
2) Solve ê problem ï terms ç F(s)
3) Transform ê soltuion F(s) back ë a solution
f(t) ç ê origïal problem
The LaPlace transform is a member ç ê INTEGRAL
TRANSFORM class ï which ê transform is given ï terms
ç an ïtegral,
èèèè ░b
F(s)è=è▒è K(s,t) f(t) dt,
èèèè ▓a
whereèK(s,t) is known as ê KERNEL ç ê transformation.
The LAPLACE TRANSFORM has as its kernel
K(s,t)è=èeúÖ▐
å is formally defïed as
èèèèèèè░▄
ÿ{ f(t) }è=è▒èeúÖ▐ f(t) dt
èèèèèèè▓╠
As is seen, this is an IMPROPER INTEGRAL because ç ê
upper limit.èIf ê function f(t) is ç EXPONENTIAL ORDER
i.e. if it is bounded by an exponential function as t ¥ ∞,
ê rapidity ç convergence çèeúÖ▐èas t ¥ ∞ will produce
a convergent ïtegral.èFor a given problem, êre may be
a limit on ê value ç constants which are forced by ê
requirement that ê ïtegral converge.
1 f(t) =è1
A)èè1 èèèB)èè1/tèèèC)è 1/sèè D)è-1/s
ü èèBy defïition
èèèèè ░▄èèèèèèèèèèè1èèè ▒bèèè1
ÿ { 1 } =è▒è1 eúÖ▐ dtè=èlim - ─── eúÖ▐ ▒è =è───
èèèèè ▓╠èèèèèèè b¥∞èèsèèè ▒╠èèès
Ç C
2 f(t) = t
A)è 1èèèèB)èè1/tèèèC)è 1/sèè D)è 1/sì
ü èè By defïition
èèèèè ░▄
ÿ { t } =è▒èt eúÖ▐ dt
èèèèè ▓╠
èèèèè
Integratïg by parts
èèèèèèèèèèèèèèèèèèèèè 1
u = tèèèdu = dtèèdv = eúÖ▐ dtè v = - ─ eúÖ▐
èèèèèèèèèèèèèèèèèèèèè s
èèèèèèèèè tèèè ▒bèè1 ░▄
ÿ { t }è=èlim - ─── eúÖ▐ ▒è + ─ ▒èeúÖ▐ dt
èèèèèèb¥∞èèsèèè ▒╠èès ▓╠
è Both limits ç ê evaluated term are zero so
èèèèèè1èèèè1èèè▒b
ÿ { t }è=è─ limè- ─ eúÖ▐ ▒
èèèèèès b¥∞èèsèèè▒╠
è The upper limit evaluates ë zero while ê lower limit
evaluates ë 1 leavïg
èèèèèè1
ÿ {t }è=è───
èèèèèès║
Ç D
3 f(t)è=ètⁿèèn a positive ïteger
A) 1/sⁿ B) 1/sⁿóî
C) n!/sⁿ D) n!/sⁿóî
üèè By defïition
èèèèèè░▄
ÿ { tⁿ } =è▒ètⁿ eúÖ▐ dt
èèèèèè▓╠èèèèè
Integratïg by parts
èèèèèèèèèèèèèèèèèèèèèè1
èèu = tⁿèèdu = ntⁿúî dtèèdv = eúÖ▐ dtè v = - ─ eúÖ▐
èèèèèèèèèèèèèèèèèèèèèès
èèèèèèèèèètⁿèèè▒bèèn ░▄
ÿ { tⁿ }è=èlim - ─── eúÖ▐ ▒è + ─ ▒ tⁿúî eúÖ▐ dt
èèèèèè b¥∞èèsèèè ▒╠èès ▓╠
è Both limits ç ê evaluated term are zero so
èèèèèè nè░▄èèè
ÿ { tⁿ }è=è─è▒ tⁿúî eúÖ▐ dt
èèèèèè sè▓╠èè
Integratïg by parts agaï
èèèèèèèèèèèèèèèèèèèèèè1
u = tⁿúîèèdu = (n-1)tⁿúì dtèèdv = eúÖ▐ dtè v = - ─ eúÖ▐
èèèèèèèèèèèèèèèèèèèèèès
èèèèèènèèèètⁿúîèèè▒bèèn n-1 ░▄
ÿ { tⁿ }è= ─èlim - ──── eúÖ▐ ▒è + ─ ─── ▒ tⁿúì eúÖ▐ dt
èèèèèèsèb¥∞èè sèèè ▒╠èèsèsè▓╠
è Both limits ç ê evaluated term are zero so
èèèèèè n(n-1)è░▄èèè
ÿ { tⁿ }è=è──────è▒ tⁿúì eúÖ▐ dt
èèèèèèès sèè▓╠èè
è Contïuïg this ïtegration by parts sequence until
ê each ïteger down ë 1 is used å ê ïtegrå is
eúÖ▐ .èOne fïal ïtegration gives aè1/sèso ê
fïal result is
èèèèèèèèèèèn (n-1) (n-2) ∙∙∙ (3)(2)(1) 1èè n!
ÿ { tⁿ }è=è──────────────────────────── ─è= ────
èèèèèèèsè sèè sè ∙∙∙èsèsèsèsèèsⁿóî
Ç D
4èèè f(t) = eì▐
èèA)è 1/ s+2èè B)è -1 / s+2èè C)è 1/ s-2è D)è-1/ s-2
üèèè By defïition
èèèèèè ░▄
ÿ { eì▐ } =è▒èeìt eúÖ▐ dt
èèèèèè ▓╠èè
è
èèèèèè ░▄
èèèèè=è▒èeúÑÖúìª▐ dt
èèèèèè ▓╠èèèèèè
Usïg subsitution
èèèèèèèèèèè1èèèèèè ▒b
èèèèè=èlimè- ───── eúÑÖúìª▐è▒
èèèèèè b¥0èè s-2èèèèèè▒╠
As long as s is greater than 2, ê function goes ë zero at
ê upper limit, leavïg only
èèèèèèè 1
èèèèè=è─────èè s > 2
èèèèèèès-2
NOTEèThis is special case ç ê general formula that
èèèèèèèèèèè1
ÿ{ e╜▐ }è= ─────èè s > a
èèèèèèèèèèèèèè s-a
ÇèC
5 f(t) = cos[2t]
èèA)è1 / sì-4èèB)è 1/ sì+4èèC)è s/ sì-4èèD)è s/ sì+4
üèè By defïition
èèèèèèèè ░▄
ÿ { cos[2t] } =è▒ècos[2t] eúÖ▐ dt
èèèèèèèè ▓╠èèèèè
Integratïg by parts
èèèèèèèèèèèèèèèèèèèèèè
èèu = cos[2t]èèdu = -2sï[2t] dtèè
èèdv = eúÖ▐ dtè v = -eúÖ▐ /s
èèèèèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèè cos[2t]èèè▒bèè2 ░▄
ÿ { cos[2t]] }è=èlim - ─────── eúÖ▐ ▒è - ─ ▒ cos[2t] eúÖ▐ dt
èèèèèèèèè b¥∞èèèsèèèè ▒╠èès ▓╠
è The first limit ç ê evaluated term is zero so
èèèèèè 1èè2è░▄èèè
ÿ { tⁿ }è=è─è- ─è▒ sï[2t] eúÖ▐ dt
èèèèèè sèèsè▓╠èè
Integratïg by parts agaï
èèèèèèèèèèèèèèèèèèèèèè
èèu = sï[2t]è du = 2cos[2t]dtè dv = eú▐ Ödtè v = -eúÖ▐ /s
èèèèèèèèèèèèèèèèèèèèèè
èèèè1èè2èèèèsï[2t]èèè▒bèè2 2 ░▄
èÿ{cos[2t]}è= ─è- ─èlim - ─────── eúÖ▐ ▒è - ─ ─ ▒ cos[2t]eúÖ▐ dt
èèèèsèèsèb¥∞èèèèsèèè ▒╠èès s ▓╠
è Both limits ç ê evaluated term are zero so
èèèè1èèè2 2 ░▄èèèèèèèèè1èè4
èÿ{cos[2t]}è= ─è -è─ ─ ▒ cos[2t]eúÖ▐ dtè= ─ - ── L{cos[2t}
èèèèsèèès s ▓╠èèèèèèèèèsè sì
Rearrangïg
(1 + 4/sì) ÿ{cos[2t]} =è1/s
orè ÿ{cos[2t]} =è1/s ÷ (1 + 4/sì)è
Invertïg, multiplyïg å simplifyïg yields
èèèèèèèès
ÿ{cos[2t]} = ───────
èèèèèèèsì+ 4
NOTE this is a special case ç
èèèèèèèès
ÿ{cos[at]} = ───────
èèèèèèèsì+aì
ÇèD
6è f(t)è=ètìe▐
èèA)è 2/(s+1)ìèèB)è2/(s+1)ÄèèC)è2/(s-1)ÄèèD)è2/(s-1)ì
üèè By defïition
èèèèèè ░▄èèèèèèèèè ░▄
ÿ{ tìe▐ } =è▒ètì e▐ eúÖ▐ dtè=è▒ètì eúÑÖúîª▐ dt
èèèèèè ▓╠èèèèèèèèè ▓╠
Integratïg by parts
èèèèèèèèèèèèèèèèèè 1
èèu = tìèèdu = 2t dtèèdv = eúÑÖúîª▐ dtè v = - ─ eúÑÖúîª▐
èèèèèèèèèèèèèèèèèèèè
èèèèèèèèèèètìèèèèè▒bèè 2è░▄
ÿ { tìe▐ }è=èlim - ─── eúÑÖúîª▐ ▒è + ─── ▒ t eúÑÖúîª▐ dt
èèèèèèè b¥∞è s-1èèèèè▒╠èès-1 ▓╠
è Both limits ç ê evaluated term are zero so
èèèèèè 2è░▄èèè
ÿ { tⁿ }è=è─è▒ t eúÑÖúîª▐ dt
èèèèèè sè▓╠èè
Integratïg by parts agaï
èèèèèèèèèèèèèèèèèèè1èèè
u = tèèdu = dtèèdv = eúÑÖúîª▐ dtè v = - ─── eúÑÖúîª▐
èèèèèèèèèèèèèèèèèè s-1
èèèèèèè2èèèè tèèèèè▒bèèè2è ░▄
ÿ { tìe▐ }è= ─èlim - ─── eúÑÖúîª▐▒è + ───── ▒èeúÑÖúîª▐ dt
èèèèèèèsèb¥∞è s-1èèèè ▒╠èè(s-1)ì▓╠
è Both limits ç ê evaluated term are zero so
èèèèèèèèè2è ░▄èèè
ÿ { tìe▐ }è=è────── ▒èeúÑÖúîª▐ dt
èèèèèèè (s-1)ì ▓╠èè
Integratïg directly
èèèèèèèè2èèèèèè1èèèèè │b
ÿ{ tìe▐ }è=è──────èlim - ─── eúÑÖúîª▐ │
èèèèèèè(s-1)ìèb¥∞è s-1èèèèè│0
èèThe value at ê upper limit goes ë zero but ê lower
limit exits å is fïite
èèèèèèèèèèè 2
ÿ{ tìe▐ } = ──────
èèèèèèèèèè(s-1)Ä
NOTE this is a special case ç
èèèèèèèèn!
ÿ{ tⁿe▐ } = ────────
èèèèèè(s-a)ⁿóî
Ç C
7èè Fïd ÿ{f»(t)}
A) ÿ{f(t)} - sf(0) B) ÿ{f(t)} + sf(0)
C) sÿ{f(t)} - f(0) D) sÿ{f(t)} + f(0)
üèè By defïition
èèèèèèè░▄èèè
ÿ{ f»(t) } =è▒èf»(t) eúÖ▐ dtè
èèèèèèè▓╠èè
è
Integratïg by parts
èèèèèèèèèèèèèèèèè
u = eúÖ▐èèdu = -seúÖ▐ dtèèdv = f»(t) dtè v = f(t)
èèèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèè ▒bèèè░▄
ÿ { f»(t) }è=èlim f(t)eúÖ▐ ▒è + s ▒ f(t)eúÖ▐ dt
èèèèèèèèb¥∞èèèèè▒╠èèè▓╠
è The upper limit ç ê evaluated term is zero while ê
lower limit evaluates ëè-f(0) .èThe ïtegral that remaïs
is ê defïition çèÿ{ f(t) } hence
ÿ{ f»(t) }è=èsÿ{ f(t) } - f(0)
ÇèC
è8 Fïdèÿ{ f»»(t) }
A) ÿ{f(t)} + sf»(0) + sìf(0)
B) ÿ{f(t)} - sf»(0) - sìf(0)
C) sìÿ{f(t)} + sf»(0) + f(0)
D) sìÿ{f(t)} - sf(0) - f»(0)
üèè By defïition
èèèèèèè ░▄èèè
ÿ{ f»»(t) } =è▒èf»»(t) eúÖ▐ dtè
èèèèèèè ▓╠èè
è
Integratïg by parts
èèèèèèèèèèèèèèèèè
u = eúÖ▐èèdu = -seúÖ▐ dtèèdv = f»»(t) dtè v = f»(t)
èèèèèèèèèèèèèèèèèèèè
èèèèèèèèèèèèèèè ▒bèèè░▄
ÿ { f»»(t) }è=èlim f»(t)eúÖ▐ ▒è + s ▒ f»(t)eúÖ▐ dt
èèèèèèèè b¥∞èèèèè ▒╠èèè▓╠
è The upper limit ç ê evaluated term is zero while ê
lower limit evaluates ëè-f»(0) .èThe ïtegral that remaïs
is ê defïition çèÿ{ f»(t) } hence
ÿ{ f»»(t) }è=è-f»(0) + sÿ{ f»(t) }
From ê last problem
ÿ{ f»(t) }è=èsÿ{ f(t) } - f(0)
So
ÿ{ f»»(t) }è=è-f»(0) + s[ sÿ{ f(t) } - f(0) ]
Rearrangïg yields
ÿ { f»»(t) }è=èsìÿ{ f(t) } - sf(0) - f»(0)
ÇèD
äèUse ê lïearity ç ê LaPlace transform ë fïd
èèèèèèèê LaPlace transform ç ê given functions
â èè Fïd ÿ{3t - 5} given that ÿ{tⁿ} = n!/sⁿóî.
By ê lïearity ç ê LaPlace transform
è ÿ{3t - 5}è=è3ÿ{t} - 5ÿ{1}
èèèèèèèèèè 1èèè 1èèè 3èè 5
èèèèèèè=è3 ──── - 5 ───è= ──── - ───
èèèèèèèèèèèèèèsìèèè sèèèsìèè s
éS èèThe LaPlace transform is a LINEAR OPERATOR ï that
èÿ{ C¬f¬(t) + C½f½(t) }è= C¬ÿ{ f¬(t) } + C½ÿ{ f½(t) }
To prove this assertion, ê defïition ç ê left hå side
is
èèèèèèèèèèèè░▄
ÿ{C¬f¬(t) + C½f½(t)} = ▒ [ C¬f¬(t) + C½f½(t) ] eúÖ▐ dt
èèèèèèèèèèèè▓╠
By ê lïearity ç ê ïtegral
èèèèèèèèèèè ░▄èèèèèèèèèè░▄
èèèèèèèè =èC¬ │ f¬(t)eúÖ▐ dtè+èC½ ▒ f½(t)eúÖ▐ dt
èèèèèèèèèèè ▓╠èèèèèèèèèè▓╠
èèèèèèèè =èC¬ÿ{f¬(t)} + C½ÿ{f½(t)}
To solve Initial Value Problems usïg ê LaPlace Transform
we will need a table.èMake a copy ç ê followïg table for
use ï ê followïg problems å ï ê next two sections.
èèèèèèèèèè1
1.èèèÿ{ 1 }è=è───
èèèèèèèèèès
èèèèèèèèèè n!
2.èèèÿ{ tⁿ } =è──────
èèèèèèèèèèsⁿóî
èèèèèèèèèè 1
3.èèèÿ{ e╜▐ } = ─────
èèèèèèèèèès-a
èèèèèèèèèèèès
4.èèèÿ{cos[at]} = ───────
èèèèèèèèèèèsì+aì
èèèèèèèèèèèèa
5. ÿ{sï[at]} = ───────
èèèèèèèèèèèsì+aì
èèèèèèèèèèèè s
6. ÿ{cosh[at]} = ───────
èèèèèèèèèèè sì-aì
èèèèèèèèèèèè a
7. ÿ{sïh[at]} = ───────
èèèèèèèèèèè sì-aì
èèèèèèèèèèèèèès-a
8. ÿ{e╜▐cos[bt]} = ───────────
èèèèèèèèèèèè (s-a)ì+bì
èèèèèèèèèèèèèè b
9. ÿ{e╜▐sï[bt]} = ───────────
èèèèèèèèèèèè (s-a)ì+bì
10. ÿ{fÑⁿª(t)} = sⁿÿ{f(t)} - sⁿúîf(0) - ∙∙∙
èèèèèèèèèèèè- sfÑⁿú²ª(0) - fÑⁿúîª(0)
9è Fïd ÿ[tì-3t+4}ègivenèÿ{tⁿ} = n!/sⁿóî
A) (2sì-3s+4)/sÄ B) (2-3s+4sì)/sÄ
C) (2sì-3s+4)/sì D) (2-3s+4sì)/sì
ü èè By lïearity
ÿ{tì-3t+4}è=èÿ{tì} -3ÿ{t} + 4ÿ{1}
èèèèèèèè2èèè 1èèè1
èèèèèè=è─── - 3──── + 4───
èèèèèèèèsÄèè sìèèès
Simplifyïg å gettïg a common denomïaër
èèèèèèèè2 - 3s + 4sì
èèèèèè=è──────────────
èèèèèèèèèè sÄ
Ç B
10 ÿ{cosh[t]}èègivenèÿ{e╜▐} = 1/ s-a
A) s/ sì+1 B) 1/ sì+1
C) s/ sì-1 D) 1/ sì-1
ü èèRecallèèèèèèe▐ + eú▐
èèèèè cosh[t] = ──────────
èèèèèèèè 2
Soèèèèèèèèè1èèèè 1
èè ÿ{cosh[t]}è=è─ ÿ{e▐} + ─ ÿ{eú▐}
èèèèèèèèèè2èèèè 2
èèèèèèèèèè1è 1èè 1è 1
èèèèèèèè =è─ ───── + ─ ─────
èèèèèèèèèè2ès-1èè2ès+1
Rearrangïg å gettïg a common denomïaër
èèèèèèèèèè s+1 + s-1
èèèèèèèè =è────────────
èèèèèèèèèè 2(s-1)(s+1)
èèèèèèèèèèè s
èèèèèèèè =è──────
èèèèèèèèèè sì-1
ÇèC