QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:gabrielmusyoka940@gmail.com
, lOMoARcPSD|44660598
CONTENTS Page
MODULE 4
LEARNING UNIT 1: LAPLACE TRANSFORMS
1.1 THE LAPLACE TRANSFORM 2
1.1.1 Definition 2
1.2 PROPERTIES OF THE LAPLACE TRANSFORM 4
1.2.1 Linearity property 4
1.2.2 First translation or shifting property 5
1.2.3 Second translation or shifting property 6
1.2.4 Change of scale property 6
1.2.5 Multiplication by t n 7
1.2.6 Division by t 8
1.3 LAPLACE TRANSFORM OF DERIVATIVES 10
1.4 LAPLACE TRANSFORM OF INTEGRALS 11
1.5 TABLE OF STANDARD LAPLACE TRANSFORMS 13
1.6 POST-TEST 14
LEARNING UNIT 2: INVERSE LAPLACE TRANSFORMS
2.1 THE INVERSE LAPLACE TRANSFORM 16
2.2 PROPERTIES OF THE INVERSE LAPLACE TRANSFORM 18
2.2.1 Linearity property 18
2.2.2 First translation or shifting property 19
2.2.3 Second translation or shifting property 19
2.2.4 Change of scale 20
2.2.5 Multiplication of s n 20
2.2.6 Division by s 21
2.2.7 The Convolution Property 22
2.3 INVERSE LAPLACE TRANSFORMS OF DERIVATIVES 23
2.4 INVERSE LAPLACE TRANSFORMS OF INTEGRALS 24
2.5 POST-TEST 34
MAT3700 iii
, lOMoARcPSD|44660598
LEARNING UNIT 3: SPECIAL FUNCTIONS
3.1 THE HEAVISIDE UNIT STEP FUNCTION
37
3.2 LAPLACE TRANSFORM OF THE HEAVISIDE UNIT STEP FUNCTION
38
3.3 THE UNIT IMPULSE FUNCTION 41
3.4 LAPLACE TRANSFORM OF THE UNIT IMPULSE FUNCTION
43
3.5 TABLE OF LAPLACE TRANSFORMS OF SPECIAL FUNCTIONS 47
3.6 POST-TEST 48
LEARNING UNIT 4: SOLVING DIFFERENTIAL EQUATIONS USING LAPLACE
TRANSFORMS
4.1 SOLUTION OF DIFFERENTIAL EQUATIONS USING
LAPLACE TRANSFORMS 50
4.2 POST-TEST 61
LEARNING UNIT 5: SOLVING SIMULTANEOUS DIFFERENTIAL EQUATIONS
USING LAPLACE TRANSFORMS
5.1 SOLUTION OF SIMULTANEOUS DIFFERENTIAL EQUATIONS USING
LAPLACE TRANSFORMS 63
5.2 POST-TEST 64
LEARNING UNIT 6: PRACTICAL APPLICATIONS
6.1 PRACTICAL APPLICATIONS OF THE LAPLACE TRANSFORM
66
6.1.1 Application to electric circuits 66
6.1.2 Application to vibrating systems 70
6.1.3 Application to beams 74
6.2 POST-TEST 83
LEARNING UNIT 2: EIGENVALUES AND EIGENVECTORS
2.1 INTRODUCTION 91
2.2 EIGENVALUES 91
2.3 EIGENVECTORS 92
2.4 POST-TEST 96
MODULE 6
LEARNING UNIT 1: FOURIER SERIES FOR PERIODIC FUNCTIONS OF
PERIOD 2π
1.1 INTRODUCTION 98
1.2 PERIODIC FUNCTIONS 98
1.3 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD 2π 99
1.4 POST-TEST 103
LEARNING UNIT 2: FOURIER SERIES FOR NON-PERIODIC FUNCTIONS OVER
A RANGE OF 2π
2.1 INTRODUCTION 105
2.2 POST-TEST 105
LEARNING UNIT 3: EVEN AND ODD FUNCTIONS AND HALF-RANGE
FOURIER SERIES
3.1 INTRODUCTION 107
3.2 REVISION: EVEN AND ODD FUNCTIONS 107
3.3 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD 2π 109
3.4 POST-TEST 111
MAT3700 v
Downloaded by Gabriel Musyoka (gabumusyoka928@gmail.com)
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller gabrielmusyoka940. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R46,49. You're not tied to anything after your purchase.
