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248 CHAP. 6 Laplace Transforms
6.8 Laplace Transform: General Formulas
Formula Name, Comments Sec.
Definition of Transform
Inverse Transform
Linearity 6.
s -Shifting (First Shifting Theorem) 6.
Differentiation of Function
Integration of Function
Convolution 6.
t -Shifting (Second Shifting Theorem)
Differentiation of Transform
Integration of Transform
f Periodic with Period p
Project 16
l( f )
1 e ps^
p
0
e stf ( t ) dt
l e
f ( t ) t
f
s
F ( s ) d s
l{ tf ( t )} F r( s )
l^1 { e asF ( s )} f ( t a ) u ( t a )
l{ ˛ f ( t a ) u ( t a )} e asF ( s )
l( f (^) * g ) l( f )l( g )
t
0
f ( t t) g (t) d t
( f * g )( t )
t
0
f (t) g ( t t) d t
l e
t
0
f (t) d tf
s l(^ f^ )
Á (^) f ( n 1)(0)
l( f ( n )) sn l( f ) s ( n 1) f (0) Á
l( f s) s^2 l( f ) sf (0) f r(0)
l( f r) s l( f ) f (0)
l^1 { F ( s a )} eatf ( t )
l{ eatf ( t )} F ( s a )
l{ af ( t ) bg ( t )} a l{ f ( t )} b l{ g ( t )}
f ( t ) l^1 { F ( s )}
F ( s ) l{ f ( t )}
0
e stf ( t ) dt
SEC. 6.9 Table of Laplace Transforms 249
6.9 Table of Laplace Transforms
For more extensive tables, see Ref. [A9] in Appendix 1.
Sec.
1 1 2 t 3 4 5 6 7 8 9
v^3
(v t sin v t )
s^2 ( s^2 v^2 )
v^2
(1 cos v t )
s ( s^2 v^2 )
eat^ cos v t s a ( s a )^2 v^2
v e
(^1) at (^) sinh v t ( s a )^2 v^2
cosh at s s^2 a^2
a sinh^ at
s^2 a^2
cos v t
s s^2 v^2
v sin^ v t
s^2 v^2
a b
( aeat^ bebt ) s ( s a )( s b )
( a b )
a b
( eat^ ebt )
( s a )( s b )
( a b )
( k )
t k ^1 eat
( s a ) k^
( k 0)
( n 1)!
t n ^1 eat
( s a ) n^
( n 1, 2, Á^ )
teat
( s a )^2
eat
s a
1 > sa^ ( a 0) t a ^1 >( a )
1 > s^3 >^221 t >p
1 > 1 s 1 > 1 p t
1 > sn^ ( n 1, 2, Á^ ) t n ^1 >( n 1)!
1 > s^2
1 > s
F ( s ) l{ ˛ f ( t )} f ( t )
(continued )
t 6.
t 6.
t 6.
x 6.
Table of Laplace Transforms (continued )
Sec.
App. Si( t ) A3.
s arccot^ s
t arctan sin v t
v s
t ln (1 cosh at )
s^2 a^2 s^2
t ln (1 cos v t )
s^2 v^2 s^2
t ln ( ebt^ eat )
s a s b
ln t g (g 0.5772) g 5.
s ln^ s
F ( s ) l{ ˛ f ( t )} f ( t )
Chapter 6 Review Questions and Problems 251
1. State the Laplace transforms of a few simple functions from memory. 2. What are the steps of solving an ODE by the Laplace transform? 3. In what cases of solving ODEs is the present method preferable to that in Chap. 2? 4. What property of the Laplace transform is crucial in solving ODEs? 5. Is? ? Explain. 6. When and how do you use the unit step function and Dirac’s delta? 7. If you know , how would you find ? 8. Explain the use of the two shifting theorems from memory. 9. Can a discontinuous function have a Laplace transform? Give reason. 10. If two different continuous functions have transforms, the latter are different. Why is this practically important? 11–19 LAPLACE TRANSFORMS Find the transform, indicating the method used and showing the details. 11. 12. 13. sin^2 (^12 p t ) 14. 16 t^2 u ( t 14 )
5 cosh 2 t 3 sinh t e t (cos 4 t 2 sin 4 t )
l^1 { F ( s )> s^2 }
f ( t ) l^1 { F ( s )}
l{ f ( t ) g ( t )} l{ f ( t )}l{ g ( t )}
l{ f ( t ) g ( t )} l{ f ( t )} l{ g ( t )}
**15. 16.
19.**
20–28 INVERSE LAPLACE TRANSFORM Find the inverse transform, indicating the method used and showing the details:
**20. 21.
- 27.**
28.
29–37 ODEs AND SYSTEMS Solve by the Laplace transform, showing the details and graphing the solution: 29.
30. y s 16 y 4 d( t p), y (0) 1, y r(0) 0
y s 4 y r 5 y 50 t , y (0) 5, y r(0) 5
3 s s^2 2 s 2
3 s 4 s^2 4 s 5
2 s 10 s^3
e ^5 s
6( s 1) s^4
s^2 6. ( s^2 6.25)^2
v cos u s sin u s^2 v^2
1 16 s^2 s (^12)
s 1 s^2
e s
s^2 2 s 8
12 t (^) * e ^3 t
t cos t sin t (sin v t ) * (cos v t )
et >^2 u ( t 3) u ( t 2 p) sin t
C H A P T E R 6 R E V I E W Q U E S T I O N S A N D P R O B L E M S
31.
32.
33. 34.
35.
36.
37.
38–45 MASS–SPRING SYSTEMS, CIRCUITS,
NETWORKS
Model and solve by the Laplace transform:
38. Show that the model of the mechanical system in Fig. 149 (no friction, no damping) is
Fig. 149. System in Problems 38 and 39
39. In Prob. 38, let . Find the solution satisfying the ini- tial conditions . 40. Find the model (the system of ODEs) in Prob. 38 extended by adding another mass and another spring of modulus in series. 41. Find the current in the RC -circuit in Fig. 150, where if if and the initial charge on the capacitor is 0.
Fig. 150. RC -circuit
R C
v ( t )
v ( t ) 40 V t 4,
R 10 , C 0.1 F, v ( t ) 10 t V 0 t 4,
i ( t )
k 4
m 3
y 2 r(0) 1 meter>sec
y 1 (0) y 2 (0) 0, y 1 r(0) 1 meter>sec,
k 2 ^ 40 kg>sec^2
m 1 m 2 10 kg, k 1 k 3 20 kg>sec^2 ,
(^0) y 1 k 1 k 2 k 3
(^0) y 2
m 2 ˛˛ y 2 s k 2 (˛˛ y 2 y 1 ) k 3 ˛ y 2 ).
m 1 ˛˛ y 1 s k 1 ˛˛˛ y 1 k 2 (˛˛ y 2 y 1 )
y 1 (0) 1, y 2 (0) 0
y 1 r y 2 u ( t p), y 2 r y 1 u ( t 2 p),
y 2 (0) 4
y 1 r 2 y 1 4 y 2 , y 2 r y 1 2 y 2 , y 1 (0) 4,
y 2 (0) 0
y 1 r 2 y 1 4 y 2 , y 2 r y 1 3 y 2 , y 1 (0) 3,
y 2 (0) 0
y 1 r y 2 , y 2 r 4 y 1 d( t p), y 1 (0) 0,
y s 3 y r 2 y 2 u ( t 2), y (0) 0, y r(0) 0
y r(0) 0
y s 4 y d( t p) d( t 2 p), y (0) 1,
y r(0) 1
y s y r 2 y 12 u ( t p) sin t , y (0) 1,
252 CHAP. 6 Laplace Transforms
42. Find and graph the charge and the current in the LC -circuit in Fig. 151, assuming if if , and zero initial current and charge. 43. Find the current in the RLC -circuit in Fig. 152, where
and current and charge at are zero.
Fig. 151. LC -circuit Fig. 152. RLC -circuit
44. Show that, by Kirchhoff’s Voltage Law (Sec. 2.9), the currents in the network in Fig. 153 are obtained from the system
Solve this system, assuming that , .
Fig. 153. Network in Problem 44
45. Set up the model of the network in Fig. 154 and find the solution, assuming that all charges and currents are 0 when the switch is closed at. Find the limits of and as , (i) from the solution, (ii) directly from the given network.
Fig. 154. Network in Problem 45
L = 5 H
Switch
C = 0.05 F
i 1 i 2 V
i 1 ( t ) i 2 ( t ) t :
t 0
v ( t )
L
R C
i 1 i 2
C 0.05 F, v 20 V, i 1 (0) 0, i 2 (0) 2 A
R 10 , L 20 H
R ( i 2 r i 1 r)
1 C i 2 0.
Li 1 r R ( i 1 i 2 ) v ( t )
R
C
L
v ( t )
C L
v ( t )
t 0
R 160 , L 20 H, C 0.002 F, v ( t ) 37 sin 10 t V,
i ( t )
v ( t ) 1 e t 0 t p, v ( t ) 0 t p
L 1 H, C 1 F,
q ( t ) i ( t )
