DEPARTMENT OF MATHEMATICAL SCIENCES FACULTY OF SCIENCE UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1793 DIFFERENTIAL EQUATIONS 1. TUTORIAL 3

Use the deﬁnition of Laplace transform to determine F (s) for the following functions. a. f (t) = 5e5t . c. f (t) = sinh 4t. e. f (t) = g. f (t) = t, 5, 0 4. t e , 0 < t < 2 h. f (t) = 0, 2 < t < 4 5, t > 4. f. f (t) =

sin 2t, 0 < t < π 0, t > π.

2.

Use the Laplace transform table to ﬁnd F (s) for the given function. a. f (t) = 2 sin t + 3 cos 2t. c. f (t) = 2t2 − 3t + 4. e. f (t) = e−2t sin 5t. g. f (t) = 5e2t + 7e−t . i. f (t) = t2 − t sinh t − 2e−t sin 3t. k. f (t) = te−t cos 2t. b. f (t) = e2t sinh2 t. d. f (t) = (sin t + cos t)2 . f. f (t) = sin t cos t. h. f (t) = (t − 1)2 + t sinh 2t. j. f (t) = t3 e−4t + t sin t. l. f (t) = 2t2 e−t cosh t.

3.

Sketch the graph of the given function for t ≥ 0, and ﬁnd its Laplace transform. a. f (t) = (t − 4)H(t − 4). c. f (t) = (t − 3)H(t − 1). e. f (t) = e−2t H(t − 4). b. f (t) = H(t − 2) − H(t − 3). d. f (t) = cos(t − π)H(t − π).

4.

Express the given function in terms of unit step functions, and ﬁnd its Laplace transform. 0, 0 < t < 2 t e, 0 < t < 2π t, 2 < t < 5 a. f (t) = b. f (t) = cos t, t > 2π. 2t e , t > 5. 0, 0

...Chapter 7
LaplaceTransform
The Laplacetransform 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 Laplacetransform 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...

...Sin Nombre
Sin Nombre, loosely translated as “without name”, is an independent film released in 2009 under the skillful direction of Cary Fukunaga. Fukunaga, a film graduate from New York University, also attended a French university and carries a bachelor’s degree in history from the University of California at Santa Cruz. During his studies and New York University, he made a short film titled Victoria Para Chino, a film about a group of immigrants who died in a refrigerated trailer when immigrating to America; The inspiration behind Sin Nombre came from that short film. In his first major production, Fukunaga continued his interest in the topic of immigration, and came up with the creation of Sin Nombre. The film follows both a young gangster of the Mara Salvatrucha gang, Casper, and young girl from Honduras, Sayra, on their difficult journey to America. Fukunaga’s overall reason for the film was to express the hardships Central American people face on their journey to America, in hopes that people could see immigration from a different light. The film is directed mainly towards citizens of America, Central America, and Mexico although it can spread to any area with controversial opinions of immigration. The constraints of the film include time, as the film lasted just 96 minutes, rating, the limited budget of an independent film, the dangerous filming locations in Central America and Mexico, and language— the...

...The Matrix and Philosophy
Have you ever felt like the life you are living right now is not real? You do
not know what is real and what is not real? You doubt your very existence. These
were main problems for Neo, the main character of the movie “The Matrix”. Neo
thought he was living a normal life, but he felt like something was wrong, he did
not know what it was, could not explain it, but something was wrong. Later on
Neo learns the truth from a man named Morpheus; he found out that what he
thought was real was actually not real at all, it was all a computer program. The
life he lived was all a lie because of his perception blocking out reality. “The
Matrix” can be compared to ontology, the study of being or what is, “The Matrix”
relates to both dualism, both body and mind are affected, and finally it relates to
the brain in a vat and the “evil genius” ruining his view of life.
Parmenides was a philosopher who believed that all reality is one, it is not a
plurality, and reality is being. He came up with “Ontology” or in Greek
“Ontos”, ontology is the study of being or “what is”. Ontology has two parts to it
one, what is, is and two, what is not is not, which is not being. The universe
consists of one thing, it never changes, it has no parts, and it can never be
destroyed, all this being one. Ontology is metaphysics. “Physics is concerned with
the microscopic processes that underlie macroscopic reality; metaphysics is
concerned with...

...Unit 5 Marketing Travel and Tourism products and services
The first pieces of work require for Unit 5 is going to ask for a definition and the function of marketing.
Advertising ,promotion products profits
Definitions of Marketing
The management process responsible for identifying, anticipating and satisfying customer requirements profitably.
(source: www.cim.co.uk)
This is all very well for privately owned businesses. Businesses have two different of section.
a type of Businesses are non-profit organisation which like public sector and voluntary, some of the building and services are provided by government, and someone who working within are also paid from government. They are also providing good services, but not looking for making money. However, the money that government had been given was came from taxes. (E.g. library, college, hoplites etc.)
Another type of businesses are privacy own which use their own money to create their own business ,providing good services and looking for making money .And all the staff are paid from their boss.
Function of Marketing
Unit 5 Marketing Travel and Tourism products and services
The first pieces of work require for Unit 5 is going to ask for a definition and the function of marketing.
Advertising ,promotion products profits
Definitions of Marketing
The management process responsible for identifying, anticipating and satisfying customer requirements profitably.
(source: www.cim.co.uk)
This is all very...

...f ( t ) = L -1 {F ( s )} 1. 3. 5. 7. 9. 11. 1 t n , n = 1, 2,3,K tsin ( at ) tsin ( at ) sin ( at ) - at cos ( at ) cos ( at ) - at sin ( at ) sin ( at + b ) sinh ( at ) e at sin ( bt ) e at sinh ( bt ) t ne at , n = 1, 2,3,K uc ( t ) = u ( t - c )
Heaviside Function
F ( s ) = L { f ( t )} 1 s n! s n +1
Table of LaplaceTransforms
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 ) tcos ( 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...

...Laplace Transformation Laplace transformation is a Mathematical tool which can be used to solve several problems in science and engineering. The transformed was first introduced by Pierre-Simon Laplace a French Mathematician, in the year 1790 in his work on probability theorem. Application of LaplaceTransform The Laplacetransform technique is applicable in many fields of science and technology such as: Control Engineering Communication Signal Analysis and Design Image Processing System Analysis Solving Differential Equations (ordinary and partial)
Advantages of Laplace transformation A Laplace transformation technique reduces the solutions of an ordinary differential equation to the solution of an algebraic equation. When the Laplacetransform technique is applied to a PDE, it reduces the number of independent variable by one. With application of Laplacetransform, particular solution of differential equation is obtained directly without necessity of first determining general solution.
Periodic Function
A real valued function ������(������) is said to be periodic with period ������ > 0 if for all ������, ������ ������ + ������ = ������(������) , and T is the least of such values. For example, sin ������ and cos ������...

...Q-A. Find the Laplacetransform of the following functions 1. f (t) = t − 1, 0 < t < 3; 7, t > 3. 2. f (t) = cost − 0,
2π 3
, 0 2π . 3
2π ; 3
4, 0 < t < 1; −2, 1 < t < 3; 3. f (t) = 5, t > 3. 5. f (t) = 3t3 + e−2t + t 3 7. f (t) = cos3 2t 9. f (t) = sin (3t + 5) 11. f (t) = e−3t sin2 t 13. f (t) = 7T 15. f (t) = e−3t (cos (4t) + 3 sin (4t)) 17. f (t) = teat 19. f (t) = t sin2 3t 21. f (t) = t2 e−2t cost 23. f (t) = tcos (7t + 9) 25. f (t) = 27. f (t) = sin2 tt e−tsintt
1 2 2
4
4. f (t) = 6. f (t) =
sint, 0 < t < π; 0, t > π. eat − 1 a
8. f (t) = cosh3 3t 10. f (t) = t2 sin (at) 12. f (t) = e−t cos2 t 14. f (t) = cat+b 16. f (t) = 4e−2t cosh (3t) 18. f (t) = t2...

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