Ordinary Differential Equations
                                                         Peter Philip∗
                                                         Lecture Notes
                       Originally Created for the Class of Spring Semester 2012 at LMU Munich,
                                   Revised and Extended for Several Subsequent Classes
                                                    September 17, 2022
                Contents
                1 Basic Notions                                                                                 4
                    1.1   Types and First Examples . . . . . . . . . . . . . . . . . . . . . . . . . .          4
                    1.2   Equivalent Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . .       10
                    1.3   Patching and Time Reversion . . . . . . . . . . . . . . . . . . . . . . . .          11
                2 Elementary Solution Methods                                                                  13
                    2.1   Geometric Interpretation, Graphing . . . . . . . . . . . . . . . . . . . . .         13
                    2.2   Linear ODE, Variation of Constants . . . . . . . . . . . . . . . . . . . . .         13
                    2.3   Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . .      16
                    2.4   Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      20
                3 General Theory                                                                               26
                    3.1   Equivalence Between Higher-Order ODE and Systems of First-Order ODE 26
                    3.2   Existence of Solutions     . . . . . . . . . . . . . . . . . . . . . . . . . . . .   29
                   ∗E-Mail: philip@math.lmu.de
                                                                1
                CONTENTS                                                                                        2
                    3.3   Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .      38
                    3.4   Extension of Solutions, Maximal Solutions . . . . . . . . . . . . . . . . .          44
                    3.5   Continuity in Initial Conditions . . . . . . . . . . . . . . . . . . . . . . .       56
                4 Linear ODE                                                                                   63
                    4.1   Definition, Setting     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   63
                    4.2   Gronwall’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      63
                    4.3   Existence, Uniqueness, Vector Space of Solutions . . . . . . . . . . . . . .         68
                    4.4   Fundamental Matrix Solutions and Variation of Constants . . . . . . . .              70
                    4.5   Higher-Order, Wronskian . . . . . . . . . . . . . . . . . . . . . . . . . . .        72
                    4.6   Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        75
                          4.6.1   Linear ODE of Higher Order . . . . . . . . . . . . . . . . . . . . .         75
                          4.6.2   Systems of First-Order Linear ODE . . . . . . . . . . . . . . . . .          83
                5 Stability                                                                                    94
                    5.1   Qualitative Theory, Phase Portraits . . . . . . . . . . . . . . . . . . . . .        94
                    5.2   Stability at Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
                    5.3   Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
                    5.4   Linearization    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
                    5.5   Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
                A Bernoulli Differential Equations                                                            132
                    A.1 α<0Is Not an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
                    A.2 α>0Is Not an Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
                    A.3 α Is an Odd Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
                    A.4 α Is an Even Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
                B Differentiability                                                                           141
                C Kn-Valued Integration                                                                      141
                    C.1 Kn-Valued Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . 142
              CONTENTS                                                                       3
                 C.2 Kn-Valued Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . 145
              D Local Lipschitz Continuity                                                 148
              E Maximal Solutions on Nonopen Intervals                                     151
              F Paths in Rn                                                                151
              G Matrix-Valued Functions                                                    155
                 G.1 Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
                 G.2 Integration and Matrix Multiplication Commute . . . . . . . . . . . . . . 156
              H Autonomous ODE                                                             157
                 H.1 Equivalence of Autonomous and Nonautonomous ODE . . . . . . . . . . 157
                 H.2 Integral for ODE with Discontinuous Right-Hand Side  . . . . . . . . . . 158
              I  Polar Coordinates                                                         159
              J Perspective: Short Introduction to PDE                                     167
                 J.1  Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
                 J.2  Laplace’s and Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . . 169
                      J.2.1 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . 169
                      J.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 171
                      J.2.3 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
                      J.2.4 Poisson’s Formula for a Ball . . . . . . . . . . . . . . . . . . . . . 174
                      J.2.5 Poisson’s Formula for a Half-Space . . . . . . . . . . . . . . . . . 174
              References                                                                   176
          1  BASIC NOTIONS                                            4
          1   Basic Notions
          1.1  Types of Ordinary Differential Equations (ODE) and First
               Examples
          Adifferential equation is an equation for some unknown function, involving one or more
          derivatives of the unknown function. Here are some first examples:
                                   y′ = y;                         (1.1a)
                                  y(5) = (y′)2 + π x;             (1.1b)
                                 (y′)2 = c;                        (1.1c)
                                  ∂ x = e2πit x2;                 (1.1d)
                                   t
                                   ′′      −1
                                  x =−3x+ 1 :                      (1.1e)
          Onedistinguishes between ordinary differential equations (ODE) and partial differential
          equations (PDE). While ODE contain only derivatives with respect to one variable,
          PDE can contain (partial) derivatives with respect to several different variables. In
          general, PDE are much harder to solve than ODE. The equations in (1.1) all are ODE,
          and only ODE are the subject of this class (a short introduction to the topic of PDE,
          including some examples, can be found in Section J of the Appendix). We will see
          precise definitions shortly, but we can already use the examples in (1.1) to get some first
          exposure to important ODE-related terms and to discuss related issues.
          As in (1.1), the notation for the unknown function varies in the literature, where the
          two variants presented in (1.1) are probably the most common ones: In the first three
          equations of (1.1), the unknown function is denoted y, usually assumed to depend on
          a variable denoted x, i.e. x 7→ y(x). In the last two equations of (1.1), the unknown
          function is denoted x, usually assumed to depend on a variable denoted t, i.e. t 7→ x(t).
          So one has to use some care due to the different roles of the symbol x. The notation
          t 7→ x(t) is typically favored in situations arising from physics applications, where t
          represents time. In this class, we will mostly use the notation x 7→ y(x).
          There is another, in a way a slightly more serious, notational issue that one commonly
          encounters when dealing with ODE: Strictly speaking, the notation in (1.1b) and (1.1d)
          is not entirely correct, as functions and function arguments are not properly distin-
          guished. Correctly written, (1.1b) and (1.1d) read
                               ∀   y(5)(x) = y′(x)
2 + πx;        (1.2a)
                              x∈D(y)
                                               

                              ∀   (∂ x)(t) = e2πit x(t) 2;        (1.2b)
                                    t
                             t∈D(x)