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Ordinary Differential Equation Lecture Notes for the Post-Graduate, Sem 3

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Ordinary Differential Equation Lecture Notes for the Post-Graduate, Sem 3 Paper XII : Ordinary Differential Equation Lecture notes for the Post-graduate, sem 3 Course Department of Mathematics Ramakrishna Mission Vidyamandira Belur Math, INDIA Course Instructor : Dr. Arnab Jyoti Das Gupta August, 2020 to January, 2021 2 Syllabus 1. Preliminaries { Initial Value problem and the equivalent integral equation, mth order equation in d-dimensions as a first order system, concepts of local existence, existence in the large and uniqueness of solutions withy examples. 2. Basic Theorems { Ascoli-Arzela Theorem. A Theorem on convergence of solutions of a family of initial-value problems. 3. Picard-Lindelof Theorem { Peano's existence Theorem and corollary. Maximal intervals of exis- tence. Extension Theorem and corollaries. Kamke's convergence Theorem. Kneser's Theorem (Statement only). 4. Differential inequalities and Uniqueness { Gronwall's inequality. Maximal and minimal solu- tions. Differential inequalities. A Theorem of Winter. Uniqueness Theorems. Nagumo's and Osgood's criteria. 5. Egres pointstand Lyapunov functions. Successive approximations. 6. Variation of constants, reduction to smaller sustems. Basic inequalities, constant coefficients. Floquet Theory. Adjoint systems, Higher order equations. 7. Linear second order equations { Preliminaries. Basic facts. Theorems of Sturm. Sturm Liou- vilee Boundary value Problems. References 1. P. Hartman, Ordinary Differential Equations, John Wiley (1964). 2. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, NY (1955). 3. G.F. Simmons : Differential Equaitons. 4. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value problems. 5. S. L. Ross, Differential Equation Contents 1 Existence and Uniqueness of solutions 5 1.1 Notations . 5 1.2 Initial Value problem . 6 1.3 Uniqueness of solutions . 16 1.3.1 Lipschitz condition . 17 1.4 Method of successive spproximations . 20 1.5 Continuation of solutions . 22 1.6 System of differential equations of first order . 24 1.7 Higher order ODEs as system of first order ODEs . 25 1.8 Dependency on Initial conditions . 28 2 System of first order ordinary differential equations 33 2.1 System of First order ODEs . 33 2.2 Systems of linear odes . 34 2.3 Uniqueness of the solution of the system of differential equations . 37 2.3.1 Existence of Fundamental set of solutions . 40 2.3.2 Linear Differential operators (constant coefficients) . 40 2.3.3 Linear Differential operators (Variable coefficients) . 44 2.3.4 Existence and uniqueness theorem . 45 2.4 Inhomogeneous system of first order linear odes . 48 2.4.1 n-th order linear ode as a system of first order linear odes . 51 2.4.2 n-th order linear ode with constant coefficients . 53 2.5 Phase potrait . 61 3 Differential Inequalities 65 3.1 Gronwall's Inequality . 65 3.2 Solution of a differential inequality . 66 4 Some more Existence and Uniqueness results 71 4.1 Maximal And Minimal solutions . 71 4.2 Uniqueness results . 73 3 4 CONTENTS 5 Sturm-Liouville Theory 79 5.1 Adjoint of a second order linear ODE . 79 5.2 Self-adjoint 2nd order linear ode . 80 5.3 Basic results of Sturm theory . 82 5.4 Sturm-Liouville Problems . 87 6 Variation of parameters 91 6.1 General theory for second order linear odes . 93 7 Liapunov functions 97 7.1 Stability of non-linear odes . 97 7.1.1 Liapunov's direct method . 98 7.2 Instability theorems . 104 Chapter 1 Existence and Uniqueness of solutions Lecture 1 1.1 Notations Through out our discussion we will be using the following notations. • I = (a; b) will denote an open interval in R. • Ck(I) will denote the set of all complex valued functions having k-continuous derivatives on I. • When I is an interval other than open interval, we can extend the above definition as follows { If f has right hand k-th derivative existing at a which is continuous from the right at a, then we will say f 2 Ck([a; b)). { If f has left hand k-th derivative existing at b which is continuous from the left at b, then we will say f 2 Ck((a; b]). { Analogously, we have the condition for f 2 Ck([a; b]). • D will denote the domain, meaning an open connected set in the real (t; x) plane, where t is the independent variable and x will be a solution or the dependent variable. • Ck(D) will denote the set of all complex valued functions on D such that all k-th order @kf partial derivatives @tp@xq ; p + q = k, exist and are continuous on D. • C0(I) or C(I) will denote the set of all continuous functions on I. • If D is such that it has multiple boundary points, which are also limit points, then one may look at the continuity of the left-hand and / or righ-hand derivatives at each such points to define Ck(D) accordingly. 5 6 CHAPTER 1. EXISTENCE AND UNIQUENESS OF SOLUTIONS Our Aim : To solve the following problem :- Find a differentiable function ' defined on I such that 1. 8t 2 I; (t; '(t)) 2 D and 2. '0(t) = f(t; '(t)); 8t 2 I. where f 2 C(D) and D is a domain. Remark. 1. Such a problem is called an ordinary differential equation of the first order. 2. It is also represented as (E) x0(t) = f(t; x); t 2 I 3. If such a differentiable function ' exists then ' is called a solution of the differential equation (E) on I. 4. Since, f 2 C(D);'0 2 C(I) =) ' 2 C1(I). 5. From the geometrical point of view, the above problem can be rephrased as finding a solution ' 2 C1(I) whose graph (t; '(t)) has slope f(t; '(t)) at the point (t; '(t)). 1.2 Initial Value problem To find an interval I containing τ and a solution ' of (E) on I satisfying '(τ) = ξ, i.e. satisfying x0(t) = f(t; x(t)); x(τ) = ξ Remark. 1. ODEs like x0(t) = 1 have infinitely many solutions x(t) = t + c, where c is a constant. 2. To avoid such situations we try to impose conditions on the solutions to obtain either a unique or a smaller class of solutions. Coming back to our initial value problem (IVP ) x0(t) = f(t; x(t)); x(τ) = ξ 1.2. INITIAL VALUE PROBLEM 7 If ' is a solution to the above problem, then we should be able to integrate both sides and obtain Z t '(t) − '(τ) = f(s; '(s))ds τ Z t =) '(t) = '(τ) + f(s; '(s))ds; 8t 2 I τ On the other hand if we start with a function implicitly defined as Z t (∗) Ψ(t) = ξ + f(s; (s))ds; 8t 2 I τ then, we have Ψ 2 C1(I) as f is continuous. Taking derivatives wrt t we have Ψ 0(t) = f(t; Ψ(t)); 8t 2 I Additionally, we have Ψ(τ) = ξ Thus, Ψ is a solution of IVP. This shows we have a one-to-one correspondence between the solutions of IVP and the C1(I) functions of the form (∗). Hence, the above IVP is equivalent to finding the solution of the integral equations (∗). Remark. 1. Though we have obtained an equivalent problem of the IVP, we have not yet solved it. 2. In fact, we still have not answered the question of whether such a solution exists or not. 3. Even if the solution exists for one particular interval I, will it exist on whole of R or on certain other intervals I0? dy 2 Example. Consider an example dt = y with the initial condition y(1) = −1. THen, clearly 1 1 y(t) = − t is a solution. But, note that t is undefined when t = 0. THus, it will be a solution only for those intervals I, that do not contain the point t = 0. Remark. The above example shows the interval plays an important role in answering the question of existence of solutions. 8 CHAPTER 1. EXISTENCE AND UNIQUENESS OF SOLUTIONS Definition 1.2.1 (-approximate solutions). Let f 2 C(D) be real valued. A function ' 2 C(I) is said to be an -approximate solution of (E) on the interval I, if it satisfies the following conditions 1. (t; '(t)) 2 D; 8t 2 I 2. ' 2 C1(I), except at most for finitely many points, where '0 may have simple discontinuities. 3. j'0(t) − f(t; '(t))j ≤ , 8t 2 InS, where S is the set of all simple discontinuities of the function '0. Remark. 1. Any function ' 2 C(I) having property (2), given above, is said to have piecewise 1 continuous derivative on I and is denoted by ' 2 Cp (I). 2. Recall that a function f has a simple discontinuity at a point c if the left and right limits of f exist at c, but are not equal. 3. If = 0, then it means ' 2 C1(I), in which case S = φ and we have our solution. Some Notations to be used later : 1. Rectangular regions R will denote the following jt − τj ≤ a; jx − ξj ≤ b; a; b > 0; i.e. R := [τ − a; τ + a] × [ξ − b; ξ + b] It is a rectangular region having center at (τ; ξ). 2. M = max jf(t; x)j, since, f is continuous and R is compact and M exists. (t;x)2R b 3. α = min a; M . Lecture 2 Theorem 1.2.2 (Existence of solution). Let f 2 C(R) and > 0. Then, there exists an -approximate solution ' of (E) on the interval [τ − α; τ + α] such that '(τ) = ξ.
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