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MAT3700 Exam pack 2024(Questions and answers)
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MAT3700 EXAMPACK 2024
QUESTIONS AND
ANSWERS
FOR ASSISTANCE CONTACT
EMAIL:gabrielmusyoka940@gmail.com
, lOMoARcPSD|31863004
LE Greyling
Department of Mathematical Sciences
UNIVERSITY OF SOUTH AFRICA
PRETORIA
, lOMoARcPSD|31863004
CONTENTS Page
LEARNING UNIT 0: INTRODUCTION vii
MODULE 1
LEARNING UNIT 1: SOLVING FIRST ORDER DIFFERENTIAL EQUATIONS
1.1 INTRODUCTION 2
1.2 FIRST ORDER DIFFERENTIAL EQUATIONS 3
1.2.1 Direct Integration- Equations without a y-term. 3
1.2.2 Equations with separable variables 4
1.2.3 Exact differential equations 5
1.2.4 Linear equations 7
1.2.5 Equations reducible to linear form - Bernoulli equations 9
1.2.6 Homogeneous equations 13
1.3 PARTICUAR SOLUTIONS 14
1.4 POST-TEST 15
LEARNING UNIT 2: PRACTICAL APPLICATIONS OF FIRST ORDER
DIFFERENTIAL EQUATIONS
2.1 GROWTH AND DECAY 19
2.2 COOLING 20
2.3 MIXTURES 22
2.4 FALLING BODIES WITH RESISTANCE 24
2.5 OTHER APPLICATIONS 26
2.6 POST-TEST 27
LEARNING UNIT 3: NUMERICAL METHODS
3.1 INTRODUCTION 30
3.2 EULER'S METHOD
30
3.3 EULER-CAUCHY METHOD 30
3.4 RUNGE-KUTTA METHOD 30
3.5 POST-TEST 31
MAT3700 iii
, lOMoARcPSD|31863004
MODULE 2
LEARNING UNIT 1: SOLVING SECOND-ORDER DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS
1.1 INTRODUCTION 33
1.2 SECOND-ORDER LINEAR EQUATIONS WITH
CONSTANT COEFFICIENTS AND Q = 0 33
1.2.1 Roots real and different 35
1.2.2 Roots real and equal 36
1.2.3 Roots complex 38
1.3 POST-TEST 41
LEARNING UNIT 2: METHOD OF UNDETERMINED COEFFICIENTS
2.1 SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT
COEFFICIENTS AND Q = f(x) OR Q = k, A CONSTANT 44
2.1.1 Q = a constant 45
2.1.2 Q = a function of the first degree in x 46
2.1.3 Q = a function of the second degree in x 47
2.1.4 Q = an exponential function 48
2.1.5 Q = a trigonometric function 50
2.1.6 Q = a hyperbolic function 51
2.1.7 Q = a sum of functions 52
2.2 PARTICULAR SOLUTIONS 53
2.3 POST-TEST 55
MODULE 3
LEARNING UNIT 1: INTRODUCTION TO DIFFERENTIAL OPERATORS
1.1 THE DIFFERENTIAL OPERATOR D 57
1
1.2 THE INVERSE OPERATOR 58
D
1.3 IMPORTANT THEOREMS 59
1.4 POST-TEST 62
MAT3700 iv
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