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Words 594

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Words 594

Pages 3

Heaviside Function

F ( s ) = L { f ( t )} 1 s n! s n +1

Table of Laplace Transforms

f ( t ) = L -1 {F ( s )}

F ( s ) = L { f ( t )} 1 s-a G ( p + 1) s p +1 1 × 3 × 5L ( 2n - 1) p 2n s 2 s 2 s + a2 s2 - a2

2 n+ 1

2. 4. 6. 8.

2

e at t p , p > -1 t n- 1 2

p

2s a 2 s + a2 2as

2

3 2

, n = 1, 2,3,K

cos ( at ) t cos ( at ) sin ( at ) + at cos ( at ) cos ( at ) + at sin ( at ) cos ( at + b ) cosh ( at ) e at cos ( bt ) e at cosh ( bt ) f ( ct )

(s

+ a2 )

10. 12.

(s

+ a2 )

2

13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37.

(s + a ) s(s - a ) (s + a )

2 2 2 2 2 2 2 2

2a 3

14. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34. 36.

(s + a ) s ( s + 3a ) (s + a )

2 2 2 2 2 2 2 2

2as 2

s sin ( b ) + a cos ( b ) s2 + a2 a 2 s - a2 b

s cos ( b ) - a sin ( b ) s2 + a2 s 2 s - a2 s-a

(s - a)

2

+ b2 -b n +1 2

(s - a)

2

+ b2 - b2

b

s-a

(s - a)

2

(s - a)

2

n!

(s - a)

1 æsö Fç ÷ c ècø e - cs e - cs L { g ( t + c )}

uc ( t ) f ( t - c ) ect f ( t ) 1 f (t ) t

e - cs s - cs e F (s) F ( s - c)

¥ s

d (t - c )

Dirac Delta Function

uc ( t ) g ( t ) t t n f ( t ) , n = 1, 2,3,K

( -1)

T 0

n

F ( n) ( s )

ò

F ( u ) du

ò f ( v ) dv

0

F (s) s

ò

t 0

f ( t - t ) g (t ) dt

F (s)G (s) sF ( s ) - f ( 0 )

f (t + T ) = f (t ) f ¢¢ ( t )

ò

e - st f ( t ) dt

f ¢ (t ) f ( n) ( t )

1 - e - sT s 2 F ( s ) - sf ( 0 ) - f ¢ ( 0 )

s n F ( s ) - s n-1 f ( 0 ) - s n- 2 f ¢ ( 0 )L - sf ( n- 2) ( 0 ) - f ( n-1) ( 0 )

Table Notes

1. This list is not a complete listing of Laplace…...

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...Laplace Transform The Laplace transform can be used to solve diﬀerential equations. Besides being a diﬀerent and eﬃcient alternative to variation of parameters and undetermined coeﬃcients, the Laplace method is particularly advantageous for input terms that are piecewise-deﬁned, periodic or impulsive. The direct Laplace transform or the Laplace integral of a function f (t) deﬁned for 0 ≤ t < ∞ is the ordinary calculus integration problem ∞ 0 f (t)e−st dt, succinctly denoted L(f (t)) in science and engineering literature. The L–notation recognizes that integration always proceeds over t = 0 to t = ∞ and that the integral involves an integrator e−st dt instead of the usual dt. These minor diﬀerences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts. 7.1 Introduction to the Laplace Method The foundation of Laplace theory is Lerch’s cancellation law ∞ −st dt 0 y(t)e = ∞ −st dt 0 f (t)e (1) L(y(t) = L(f (t)) implies or implies y(t) = f (t), y(t) = f (t). In diﬀerential equation applications, y(t) is the sought-after unknown while f (t) is an explicit expression taken from integral tables. Below, we illustrate Laplace’s method by solving the initial value problem y = −1, y(0) = 0. The method obtains a relation L(y(t)) = L(−t), whence Lerch’s cancellation law implies the solution is y(t) = −t. The Laplace method is advertised as a table lookup method, in which the solution y(t) to a diﬀerential......

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