Qingkai Kong - A Short Course in Ordinary Differential Equations

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Springer International Publishing Switzerland 2014

Qingkai Kong A Short Course in Ordinary Differential Equations Universitext 10.1007/978-3-319-11239-8_1

1. Initial Value Problems

Qingkai Kong 1

(1)

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, USA

1.1 Introduction

As you have learned from an elementary differential equations course, a differential equation, more precisely an ordinary differential equation, is an equation for an unknown function of a single variable which involves certain derivatives of the unknown function. The order of the highest derivative of the unknown function appearing in the equation is called the order of the differential equation. Based on this, n th-order scalar differential equations have the general form

(1.1.1)

where F is a function of n + 2 variables. If F satisfies certain conditions, then Eq. () can be written into the standard form

(1.1.2)

where f is a function of n + 1 variables. A solution of Eq. (), if it exists, is a solution that contains n arbitrary constants . Under certain assumptions on the function F or f , the values of can be uniquely determined for a particular solution by n initial conditions (ICs) defined at a single point (initial point) t 0:

(1.1.3)

where . The problem consisting of Eq. () is called an initial value problem (IVP) .

The above concepts for scalar differential equations and IVPs can be extended to vector-valued differential equations and IVPs. For instance, the standard form of first-order systems of differential equations is

(1.1.4)

where and . A solution of system (), if it exists, is a solution that contains n arbitrary constants . Under certain assumptions on the function f , the values of can be uniquely determined for a particular solution by n ICs

(1.1.5)

where and . The problem consisting of system () is an IVP.

We comment that any standard scalar or vector-valued differential equation or IVP can be changed to a first-order system of differential equations or IVP. For example, if we let

then Eq. () becomes the system of differential equations

and IC ().

Under what conditions does IVP () have a unique solution? Before answering this question, lets look at the following examples of scalar IVPs.

Example 1.1.1.

Consider the IVP

(1.1.6)

By solving the problem we see that the solution of IVP () is which exists on the whole real number line and is unique.

Example 1.1.2.

Consider the IVP

(1.1.7)

By solving the problem we see that when x 0 0, IVP (.

Fig. 1.1

Solution may not exist on the whole line

Example 1.1.3.

Consider the IVP

(1.1.8)

Clearly, x = 0 is a solution of IVP () for t [ c , ). By combining this solution with the zero solution, we have that for any c > 0

is also a solution of IVP () which exists on . Similarly, for any c < 0

is also a solution of IVP (.

Fig. 1.2

Nonunique solutions

Example 1.1.4.

Consider the IVP

(1.1.9)

We claim that IVP () has no solution at all. In fact, if it had a solution, then the solution would be x = | t |. However, this function is not differentiable at t = 0. We have reached a contradiction.

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