1 Ordinary differential equations

Introduction

This chapter begins with the well-known formula (see, e.?g., Apostol's book Calculus [4])

giving the explicit solutions of the linear equation x´=a(t)x+b(t), and one of its targets is to lead the reader to the extension of (1.0.1) to first-order linear systems such as

where A,B are, respectively, an n×n matrix and an n-column vector with real continuous elements. As will be clear from the discussion in Section 1.3, this generalization is based on the fusion (if we can say so) between some well-known and basic facts from linear algebra on the one hand and the (global) existence theorem for the Cauchy problem attached to (1.0.2) (see Lemma 1.3.1) on the other hand.

In this presentation, two intermediate steps need to be made for the described extension of (1.0.1) to (1.0.2). The first consists in passing from the linear equation x´=a(t)x+b(t) to the general first-order equation (in normal form)

We do this in Section 1.1 with a quick look at the main existence and uniqueness theorems concerning the initial value problem (IVP) for (1.0.3); practically no proofs will be given for the statements involved, and we refer since this very beginning to for instance the books of Hale [5] or Walter [1] for an adequate treatment of this topic. The second intermediate step consists of course in passing from the first-order equation (1.0.3) to first-order general systems X´=F(t,X), and this is sketched in a straightforward way in Section 1.2.

Section 1.4 deals with the important case in which the matrix A in (1.0.2) is independent from t (linear systems with constant coefficients) and gives the opportunity to discuss a nice and conceptually important extension of the familiar Taylor series expansion

of the exponential function of the real variable x to the case where x is replaced by an n-by-n matrix A.

The final Sections 1.5 and 1.6 of this chapter deal with higher-order differential equations, with the declared target of gaining some familiarity with second-order linear equations. Giving special importance to the second-order equation

is justified not only by its importance in physics coming from the fundamental principle of dynamics

and thus from the classical paradigm stating the uniqueness of the motion of a particle with given initial position and velocity, but also by the fact that (1.0.4) are the simplest to examine from the point of view of boundary value problems (BVPs), in which the initial conditions on x and x´ at a given point t0 belonging to the interval I=[a,b] (assuming f is defined on I×R2) are replaced by conditions involving x and/or x´ at the endpoints a and b of the interval. Such kind of problems (i) yield a good starting point for the study of nonlinear operator equations and (ii) give rise to the classical theory of Sturm and Liouville for linear equations, which will be cited with some details in the Additions to Chapter 3. Finally, they serve as an introduction to the much more difficult BVPs for second-order partial differential equations (PDEs), in particular those of mathematical physics, which we shall synthetically discuss in Chapter 4.

1.1 Ordinary differential equations (ODEs) of the first order

Example 1.1.1.

The first-order linear equation

with coefficients a,b?C(I), can be solved by means of the explicit formula

where for f?C(I), the symbol ?f(t)dt denotes an arbitrarily chosen primitive of f in I. For instance, the equation

has the solutions x(t)=c/t+t/2,c?R.

Example 1.1.2.

Some simple nonlinear equations can also be solved explicitly. For instance, consider

where h?C(B), B?R. If along a solution x(t) we have h(x(t))?0, then x´(t)/h(x(t))=a(t), and integrating both members of this equality gives x(t) itself if one is able to find a primitive of 1/h. For instance, the equations

can be solved to yield respectively

with k?R. The three equations all have in addition the solution x=0; moreover, (iii) has a second "trivial" solution, namely, x=1.

Consider now a first-order differential equation in general (normal) form:

where f=f(t,x) is a real-valued function defined in a subset A of R2. To study (1.1.3) we first need to define precisely what is a solution of it.

Definition 1.1.1.

A solution of the differential equation (1.1.3) is a function u=u(t), defined in some interval J=Ju?R, such that:

(a)

u is differentiable in J,

(b)

(t,u(t))?A for all t?J, and

(c)

u´(t)=f(t,u(t)) for all t?J.

Exercise 1.1.1.

Prove that if f is continuous on its domain A, then any solution of (1.1.3) is of class C1 on its interval of definition.

Remark on the notations.

In order to emphasize and clarify the concept of solution, we have employed in Definition 1.1.1 a different symbol for the unknown (x) of the differential equation and for a solution (u) of it. Of course, this is simply a matter of taste in the use of notations, and throughout these Notes we will ourselves use the same symbol with both meanings (as already done in the examples displayed above).

These examples suggest that we have to expect infinitely many solutions of an equation like (1.1.3). A fundamental remark is that in Example 1.1.1, there is exactly one solution satisfying a given "initial" condition of the form x(t0)=x0: just impose that condition in (1.1.2) to obtain uniquely c. One can check that it is the same for equations (ii) and (iii) in Example 1.1.2, but it is not the same for (i) if we take the initial condition x(t0)=0, for we have in this case the two different solutions u1(t)=(t-t0)3 and u2(t)=0.

Driven by these remarks, we shall study from now on the initial value problem (IVP), also called Cauchy problem, for the differential equation (1.1.3), which is written

and consists in finding a solution u of the differential equation, defined in an interval J containing the point t0 and such that...