Introduction

In this chapter we illustrate the qualitative approach to differential equations and introduce some key ideas such as phase portraits and qualitative equivalence.

1.1 PRELIMINARY IDEAS

1.1.1 Existence and uniqueness

Definition 1.1 J Let X (t, x) be a real-valued function of the real variables t and x, with domain D ^ OS2. A function x(t), with t in some open interval / c R , which satisfies

*(t) = ^ = X(t,x(t)) (1.1)

is said to be a solution of the differential equation (1.1). A necessary condition for x(t) to be a solution is that (t,x(t))eD for each

tel; so that D limits the domain and range of x(t). If x(t), with domain J, is a solution to (1.1) then so is its restriction to any interval J cz 1. To prevent any confusion, we will always take 1 to be the largest interval for which x(t) satisfies (1.1). Solutions with this property are called maximal solutions. Thus, unless otherwise stated, we will use the word ‘solution’ to mean ‘maximal solution’. Consider the following examples of (1.1) and their solutions; we give

x = X(t9x), D, x(t), I

in each case (C and C are real numbers):

1. x = x — £, OS2, 1 + 1 + Ce*, OS; 2. x = x2, OS2, (-o o ,C )

3 . x = —x/£, {(t9x)\t 7*0}, C/t, (—oo,0) C /t9 (0, oo);

4. x = 2x1/2,

0, R: 5. x = 2xt, IR2, Ce*2, IR; 6. x = — x/tanh t, {(£,x)|£^0}, C/sinht, (—oo,0)

C'/sinht, (0, oo).