Ordinary Differential Equations: Study Guide for MAT3706 (UNISA)

.
Department of Mathematical Sciences
UNIVERSITY OF SOUTH AFRICA
Ordinary
Differential Equations
Only Study Guide for
.MAT3706
Author
Prof M Grobbelaar
1st Revised by
Dr SA van Aardt
2nd Revised by
Prof H Jafari

c
2018 University of South Africa
All rights reserved
Printed and Published by the
University of South Africa
Muckleneuk, Pretoria
MAT3706/1/2019

iii MAT
Contents
List of Figures iv
PREFACE 0
1 Linear Systems of Differential Equations, Solution of Systems of Differential Equations
by the Method of Elimination 1
1.1 Introduction ............................................ 2
1.2 Definitions and basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Themethodofelimination .................................... 6
1.3.1 Polynomialoperators ................................... 7
1.3.2 Equivalent triangular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Degenerate systems of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Applications............................................. 18
1.5.1 Electricalcircuits ..................................... 18
1.5.2 Mixtureproblems ..................................... 20
1.5.3 Loveaffairs ......................................... 22
2 Eigenvalues and Eigenvectors and Systems of Linear Equations with Constant Coeffi-
cients 23
2.1 Introduction ............................................ 24
2.2 The eigenvalue–eigenvector method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Linear independence of eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Complex eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 New roots of ˙ X=AX ...................................... 33
2.6 Initial value problems; solution of initial value problems by the eigenvalue–eigenvector method 36
3 Generalised Eigenvectors (Root Vectors) and Systems of Linear Differential Equations 40
3.1 Generalised eigenvectors or root vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Root vectors and solutions of ˙ X=AX ............................. 44
3.3 Two important results (and keeping our promises!) . . . . . . . . . . . . . . . . . . . . . . 48
4 Fundamental Matrices,
Non–homogeneous Systems,

iv
The Inequality of Gronwall 51
4.1 Fundamentalmatrices....................................... 52
4.2 The uniqueness theorem for linear systems with constant coefficients . . . . . . . . . . . . . 55
4.3 Applications of the uniqueness theorem, solution of the non–homogeneous problem . . . . 58
4.4 The non–homogeneous problem ˙ X=AX +F(t): Variation of parameters . . . . . . . . . . 59
4.5 TheinequalityofGronwall .................................... 67
5 Higher Order One–dimensional Equations as Systems of First Order Equations 73
5.1 Introduction ............................................ 73
5.2 Companion systems for higher order one dimensional differential equations . . . . . . . . . 73
6 Analytic Matrices and Power Series Solutions of Systems of Differential Equations 80
6.1 Introduction ............................................ 80
6.2 Power series expansions of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 SERIES SOLUTIONS FOR ˙ X=A(t)X............................ 85
6.4 Theexponentialofamatrix ................................... 91
7 Nonlinear Systems,
Existence and Uniqueness Theorem for Linear Systems 93
7.1 Nonlinear equations and systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Numerical solutions of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3 Existence and uniqueness theorem for linear systems of differential equations . . . . . . . . 98
8 Qualitative theory of ODE, stability of solutions of linear systems and linearization of
nonlinear systems 101

v MAT3706/1
List of Figures
1.1 Anelectricalcircuit ........................................ 18
1.2 Flowbetweentwotanks ...................................... 21