Existence and uniqueness theorem for ODE: an overview

📝 Abstract

The study of existence and uniqueness of solutions became important due to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years after the development of the theory of differential equation. In the article, we shall discuss briefly the differences between linear and nonlinear first order ODE in context of existence and uniqueness of solutions. Special emphasis is given on the Lipschitz continuous functions in the discussion.

💡 Analysis

The study of existence and uniqueness of solutions became important due to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years after the development of the theory of differential equation. In the article, we shall discuss briefly the differences between linear and nonlinear first order ODE in context of existence and uniqueness of solutions. Special emphasis is given on the Lipschitz continuous functions in the discussion.

📄 Content

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Existence and uniqueness theorem for ODE: an overview Swarup Poria and Aman Dhiman Department of Applied Mathematics, University of Calcutta,
92, A.P.C.Road, Kolkata-700009, India

  Abstract: The study of existence and uniqueness of solutions became important due 

to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years after the development of the theory of differential equation. In the article, we shall discuss briefly the differences between linear and nonlinear first order ODE in context of existence and uniqueness of solutions. Special emphasis is given on the Lipschitz continuous functions in the discussion.

1. Introduction: Differential equations are essential for a mathematical description of nature, many of the general laws of nature-in physics, chemistry, biology, economics and engineering –find their most natural expression in the language of differential equation. Differential Equation(DE) allows us to study all kinds of evolutionary processes with the properties of determinacy; finite-dimensionality and differentiability. The study of DE began very soon after the invention of differential and integral calculus. In 1671, Newton had laid the foundation stone for the study of differential equations. He was followed by Leibnitz who coined the name differential equation in 1676 to denote relationship between differentials 𝑑𝑥 and 𝑑𝑦 of two variables 𝑥 and 𝑦. The fundamental law of motion in mechanics, known as Newton’s second law is a differential equation to describe the state of a system. Motion of a particle of mass m moving along a straight line under the influence of a specified external force 𝐹(𝑡, 𝑥, 𝑥′) is described by the following DE 𝑚𝑥′′ = 𝐹(𝑡, 𝑥, 𝑥′) ; (𝑥′ = 𝑑𝑥 𝑑𝑡, 𝑥′′ = 𝑑2𝑥 𝑑𝑡2) (1) At early stage, mathematicians were mostly engaged in formulating differential equations and solving them but they did not worry about the existence and uniqueness of solutions.

More precisely, an equation that involves derivatives of one or more unknown dependent variables with respect to one or more independent variables is known as differential equations. An equation involving ordinary derivatives of one or more dependent variables with respect to single independent variable is called an ordinary differential equations (ODEs). A general form of an ODE containing one independent and one dependent variable is F(𝑥, 𝑦, 𝑦′, 𝑦′′, … … , 𝑦𝑛) = 0 where F is an arbitrary function of 𝑥, 𝑦, 𝑦′, … … , 𝑦𝑛, here 𝑥 is the independent variable while y being the dependent variable and 𝑦𝑛≡ 𝑑𝑛𝑦 𝑑𝑥𝑛. The order of an ODE is the order of the highest derivative appearing in it. Equation (1) is an example of second order ODE. On the other 2

hand, partial differential equations are those which have two or more independent variables. Differential equations are broadly classified into linear and non-linear type. A differential equation written in the form 𝐹(𝑥, 𝑦, 𝑦′, 𝑦′′, … … , 𝑦𝑛) = 0 is said to be linear if 𝐹 is a linear function of the variables 𝑦, 𝑦′, 𝑦′′, … … , 𝑦𝑛(except the independent variable 𝑥). Let 𝐿 be a differential operator defined as 𝐿≡𝑎𝑛(𝑥)𝐷𝑛+ 𝑎(𝑛−1)(𝑥)𝐷𝑛−1 + ⋯+ 𝑎1(𝑥)𝐷+ 𝑎0(𝑥) where 𝐷𝑛≡ 𝑑𝑛𝑦 𝑑𝑥𝑛; (for n=1,2,….).One can easily check that the operator 𝐿 satisfies the condition for linearity i.e. 𝐿(𝑎𝑦1 + 𝑏𝑦2) = 𝑎𝐿(𝑦1) + 𝑏𝐿(𝑦2) for all 𝑦1, 𝑦2 with scalars 𝑎 and 𝑏. Therefore a linear ODE of order 𝑛 can be written as 𝑎𝑛(𝑥)𝑦(𝑛) + 𝑎𝑛−1(𝑥)𝑦(𝑛−1) + ⋯+ 𝑎0(𝑥)𝑦= 𝑓(𝑥) (2)
An ODE is called nonlinear if it is not linear. In absence of damping and external driving, the motion of a simple pendulum (Fig.1) is governed by

𝑑2𝜃 𝑑𝑡2 + 𝑔 𝐿sin 𝜃= 0 (3) where 𝜃 is the angle from the downward vertical, 𝑔 is the acceleration due to gravity and 𝐿 is the length of the pendulum. This is a famous example of nonlinear ODE.
In this article, we shall confine our discussion to single scalar first order ODE only. An initial value problem(IVP) for a first order ODE is the problem of finding solution 𝑦= 𝑦(𝑥) that satisfies the initial condition 𝑦(𝑥0) = 𝑦0 where 𝑥0, 𝑦0 are some fixed values.We write the IVP of first order ODE as 𝑑𝑦 𝑑𝑥= 𝑓(𝑥, 𝑦), 𝑦(𝑥0) = 𝑦0 . (4) In other words, we intend to find a continously differentiable function 𝑦(𝑥) defined on a confined interval of 𝑥 such that 𝑑𝑦 𝑑𝑥= 𝑓(𝑥, 𝑦(𝑥)) and 𝑦(𝑥0) = 𝑦0. Such a function 𝑦= 𝑦(𝑥) is called a solution to the equation (4). In integral form 𝑦= 𝑦(𝑥) is the solution of equation (4) if it satisfies the integral equation 𝑦(𝑥) = 𝑦0 + ∫𝑓(𝑢, 𝑦(𝑢))𝑑𝑢 𝑥 𝑥0 .
There are several well known methods namely separation of variables, variation of parameters, methods using integrating fact

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