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 19, 2021 Contents 1 Basic Notions 4 1.1 TypesandFirstExamples .......................... 4 1.2 EquivalentIntegralEquation. ... 10 1.3 PatchingandTimeReversion . 11 2 Elementary Solution Methods 13 2.1 GeometricInterpretation,Graphing . ..... 13 2.2 LinearODE,VariationofConstants. ... 13 2.3 SeparationofVariables . 16 2.4 ChangeofVariables.............................. 20 3 General Theory 26 3.1 Equivalence Between Higher-Order ODE and Systems of First-Order ODE 26 3.2 ExistenceofSolutions ............................ 29 ∗E-Mail: [email protected] 1 CONTENTS 2 3.3 UniquenessofSolutions. 38 3.4 Extension of Solutions, Maximal Solutions . ...... 44 3.5 ContinuityinInitialConditions . .... 56 4 Linear ODE 63 4.1 Definition,Setting .............................. 63 4.2 Gronwall’sInequality . 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 ConstantCoefficients ............................. 75 4.6.1 LinearODEofHigherOrder. 75 4.6.2 SystemsofFirst-OrderLinearODE . 83 5 Stability 93 5.1 QualitativeTheory,PhasePortraits . ..... 93 5.2 StabilityatFixedPoints . 104 5.3 ConstantCoefficients .............................112 5.4 Linearization .................................118 5.5 LimitSets ...................................123 A Differentiability 131 B Kn-Valued Integration 132 B.1 Kn-ValuedRiemannIntegral. 133 B.2 Kn-ValuedLebesgueIntegral. 136 C Metric Spaces 139 C.1 DistanceinMetricSpaces . 139 D Local Lipschitz Continuity 142 CONTENTS 3 E MaximalSolutionsonNonopenIntervals 145 F Paths in Rn 145 G Operator Norms and Matrix Norms 149 H The Vandermonde Determinant 153 I Matrix-Valued Functions 155 I.1 ProductRule .................................155 I.2 Integration and Matrix Multiplication Commute . .......156 J Autonomous ODE 156 J.1 Equivalence of Autonomous and Nonautonomous ODE . 156 J.2 Integral for ODE with Discontinuous Right-Hand Side . ......157 K Polar Coordinates 158 References 166 1 BASIC NOTIONS 4 1 Basic Notions 1.1 Types of Ordinary Differential Equations (ODE) and First Examples A differential 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 + πx, (1.1b) (y′)2 = c, (1.1c) 2πit 2 ∂tx = e x , (1.1d) 1 x′′ = 3x + − . (1.1e) − 1 One distinguishes 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. 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 y(x). In the last two equations of (1.1), the unknown function is denoted x, usually7→ assumed to depend on a variable denoted t, i.e. t x(t). So one has to use some care due to the different roles of the symbol x. The notation7→ t x(t) is typically favored in situations arising from physics applications, where t represents7→ time. In this class, we will mostly use the notation x y(x). 7→ 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 2 y(5)(x)= y′(x) + πx, (1.2a) x∈D∀(y) 2πit 2 (∂tx)(t)= e x(t) , (1.2b) t∈D∀(x) 1 BASIC NOTIONS 5 where (y) and (x) denote the respective domains of the functions y and x. However, one mightD also noticeD that the notation in (1.2) is more cumbersome and, perhaps, harder to read. In any case, the type of slight abuse of notation present in (1.1b) and (1.1d) is so common in the literature that one will have to live with it. One speaks of first-order ODE if the equations involve only first derivatives such as in (1.1a), (1.1c), and (1.1d). Otherwise, one speaks of higher-order ODE, where the precise order is given by the highest derivative occurring in the equation, such that (1.1b) is an ODE of fifth order and (1.1e) is an ODE of second order. We will see later in Th. 3.1 that ODE of higher order can be equivalently formulated and solved as systems of ODE of first order, where systems of ODE obviously consist of several ODE to be solved simultaneously. Such a system of ODE can, equivalently, be interpreted as a single ODE in higher dimensions: For instance, (1.1e) can be seen as a single two-dimensional ODE of second order or as the system x′′ = 3x 1, (1.3a) 1 − 1 − x′′ = 3x +1 (1.3b) 2 − 2 of two one-dimensional ODE of second order. One calls an ODE explicit if it has been solved explicitly for the highest-order deriva- tive, otherwise implicit. Thus, in (1.1), all ODE except (1.1c) are explicit. In general, explicit ODE are much easier to solve than implicit ODE (which include, e.g., so-called differential-algebraic equations, cf. Ex. 1.4(g) below), and we will mostly consider ex- plicit ODE in this class. As the reader might already have noticed, without further information, none of the equations in (1.1) makes much sense. Every function, in particular, every function solving an ODE, needs a set as the domain where it is defined, and a set as the range it maps into. Thus, for each ODE, one needs to specify the admissible domains as well as the range of the unknown function. For an ODE, one usually requires a solution to be defined on a nontrivial (bounded or unbounded) interval I R. Prescribing the possible range of the solution is an integral part of setting up an ODE,⊆ and it often completely determines the ODE’s meaning and/or its solvability. For example for (1.1d), (a subset of) C is a reasonable range. Similarly, for (1.1a)–(1.1c), one can require the range to be either R or C, where requiring range R for (1.1c) immediately implies there is no solution for c < 0. However, one can also specify (a subset of) Rn or Cn, n > 1, as range for (1.1a), turning the ODE into an n-dimensional ODE (or a system of ODE), where y now has n compoments (y1,...,yn) (note that, except in cases where we are dealing with matrix multiplications, we sometimes denote elements of Rn as columns and sometimes as rows, switching back and forth without too much care). A reasonable range for (1.1e) is (a subset of) R2 or C2. 1 BASIC NOTIONS 6 One of the important goals regarding ODE is to find conditions, where one can guan- rantee the existence of solutions. Moreover, if possible, one would like to find conditions that guarantee the existence of a unique solution. Clearly, for each a R, the function y : R R, y(x)= aex, is a solution to (1.1a), showing one cannot expect∈ uniqueness without−→ specifying further requirements. The most common additional conditions that often (but not always) guarantee a unique solution are initial conditions, (e.g. requiring y(x0)= y0 (x0,y0 given); or boundary conditions (e.g. requiring y(a)= ya, y(b)= yb for y : [a,b] Cn (y ,y Cn given)). −→ a b ∈ Let us now proceed to mathematically precise definitions of the abovementioned notions. Notation 1.1. We will write K in situations, where we allow K to be R or C. Definition 1.2. Let k,n N. ∈ (a) Given U R K(k+1)n and F : U Kn, call ⊆ × −→ F (x,y,y′,...,y(k))=0 (1.4) an implicit ODE of kth order. A solution to this ODE is a k times differentiable function φ : I Kn, (1.5) −→ defined on a nontrivial (bounded or unbounded, open or closed or half-open) interval I R satisfying the two conditions ⊆ (i) (x,φ(x),φ′(x),...,φ(k)(x)) I K(k+1)n : x I U. ∈ × ∈ ⊆ (ii) F (x,φ(x),φ′(x),...,φ(k)(x)) = 0 for each x I. ∈ Note that condition (i) is necessary so that one can even formulate condition (ii). (b) Given G R Kkn, and f : G Kn, call ⊆ × −→ y(k) = f(x,y,y′,...,y(k−1)) (1.6) an explicit ODE of kth order. A solution to this ODE is a k times differentiable function φ as in (1.5), defined on a nontrivial (bounded or unbounded, open or closed or half-open) interval I R satisfying the two conditions ⊆ (i) x,φ(x),φ′(x),...,φ(k−1)(x) I Kkn : x I G. ∈ × ∈ ⊆ (ii) φ(k)(x)= f(x,φ(x),φ′(x),...,φ(k−1)(x)) for each x I. ∈ Again, note that condition (i) is necessary so that one can even formulate condition (ii). Also note that φ is a solution to (1.6) if, and only if, φ is a solution to the equivalent implicit ODE y(k) f(x,y,y′,...,y(k−1))=0.