Differential Equations

I. The existence and uniqueness theorem

1. Résumé of some elementary theory of differential equations

As has been pointed out in the preface, the theory of differential equations was for many years concerned with the following problem—given a differential equation, i.e. an equation of the form¹

(1)

it is required to integrate the equation in explicit terms, i.e. to find explicitly a function y of x and of one or more arbitrary constants C1, C2, C3, . . . ,² which when substituted into (1) satisfies it identically, and in addition is of a sufficient degree of generality.

This second condition, which as stated above is vague and requires exact definition, would however be fully satisfied were it possible to write down a formula containing all the possible solutions of (1), as, for example, may be done in the case of an equation with variables separable such as

whose solutions (usually called its integrals) are all given by the formula

where C is an arbitrary constant. (It is assumed that B(y) e9780486299297_img_8802.gif 0.)

However, as it is not always possible, or at least not always easy, to determine a formula of this kind, the concept of a general integral of a differential equation of the form (1) has gradually evolved in the development of modern analysis. At the present day this term denotes a function

(2)

of the independent variable x and of n arbitrary constants C1, C2, C3, . . ., Cn, the function y containing these constants in such a way that, by attributing suitable values to them, it is always possible to satisfy at a particular point x = x0 "initial conditions" of the form

(3)

for any prescribed y0, e9780486299297_i0007.jpg , e9780486299297_i0008.jpg , . . . , e9780486299297_i0009.jpg whatsoever.

We do not therefore exclude in any way the possible existence of further solutions of the differential equation which are not deducible from (2) by giving particular constant values to all the arbitrary constants,³ but we demand only that the general integral be sufficiently flexible to satisfy the initial conditions (3).

Knowledge of the general integral of a differential equation is generally sufficient to enable us to solve the various problems that may arise relative to the equation—for example, that of determining the solution (or solutions) of the equation satisfying additional conditions of a different type from (3). Unfortunately the explicit determination of the general integral is not possible except in relatively rare cases and sometimes, even if possible, it may give rise to formulæ so complicated as to be of little use. Modern analysis, however, although persisting in the search for the general integral of a given equation,⁴ prefers, as has already been pointed out, to study the main properties of its integrals directly, no matter whether the equation is to be studied alone or in conjunction with additional conditions of the type (3), or of another kind. We shall see this fully illustrated in the subsequent chapters.

Another concept from the elementary theory of differential equations which will be employed, in addition to that of a general integral, is that of the equivalence of a single equation and of a system of differential equations.

For example, given a differential equation of order n, which for greater simplicity we shall suppose solved for the derivative of highest order, i.e. we shall suppose it written in the form

(4)

on putting

(5)

we may write (4) as a system of n differential equations of the first order, viz.

(6)

This system is a particular case of the differential system

(7)

Systems of the form (7) are called normal systems of differential equations (of the first order).

Conversely, given a system of the form (7), by differentiating each equation (n–1) times with respect to x and eliminating from the n² equations thus obtained the n²–1 unknowns

we obtain a single equation of order n for y1 (solved for e9780486299297_i0015.jpg ), and similarly for y2, y3, . . . , yn.

Without pursuing possible easy generalizations of these last remarks—for example, their extension to a system of differential equations not all of the first order—we shall confine ourselves to the further remark that in view of (5) the initial conditions corresponding to (3) take the form

(8)

where e9780486299297_i0017.jpg , e9780486299297_i0018.jpg , . . . , e9780486299297_i0019.jpg denote arbitrary constants.

2. Preliminaries to the fundamental theorem

As first aim we wish to establish a basic fundamental theorem for the whole theory of differential equations, a theorem which asserts that, under fairly weak restrictions, a differential equation solved for the derivative of highest order possesses one and only one general integral. We thus consider, with the advantage of greater generality, a normal differential system of the form (7)⁵, and we shall show that under fairly wide conditions on the functions f1, f2, . . . , fn, there exists one and only one solution—i.e. one and only one n-ple of functions y1(x), ... , yn(x) satisfying identically the system of equations and for which the initial conditions (8) are satisfied.

We demand that the functions fi(x, y1, y2 , . . . , yn), (i = 1, 2, 3, . . . , n) be continuous with respect to the variables x, y1, . . . , yn, and, in addition, that within suitable ranges for each of the variables y1, y2, . . . , yn, they satisfy a Lipschitz condition with respect to each variable yl, y2, . . . , yn. This condition is rather more restrictive than the demand of continuity but less restrictive than that of differentiability with respect to the variable concerned. In fact, a Lipschitz condition with respect to a function F(x) in a certain interval (a, b) asserts that the relative increments of the function remain bounded in that interval, i.e. that there exists a positive constant A such that for any two points x1, x2 of the interval (a, b)

(9)

If a function of several variables F(x1, x2, . . . , xm) satisfies a Lipschitz condition with respect to each of these variables, within suitable ranges,⁶ i.e. if

(10)

then the function F satisfies a Lipschitz condition also with respect to the set of variables x1, x2, . . . , xm, in the sense that there exists a constant A such that

(11)

for we have the identity

from which, by (10), follows

and we need only take A to be the greatest of the m constants A1, A2, . . . , Am to obtain (11).

3. The existence and uniqueness theorem for normal differential systems

We can now enunciate the fundamental theorem in the following form:—

Given a differential system of the form (7) together with the initial conditions (8); if two positive numbers a and b can be found such that in the domain D defined by the inequalities⁷

(12)

the functions f1, f2, . . . , fn are continuous and satisfy Lipschitz conditions in the variables y1, y2, . . . , yn within these ranges, then a positive number δ( e9780486299297_img_8806.gif a) can be determined such that in the interval (x0, x0 + δ) the system possesses one and only one solution satisfying the initial conditions (8).

To prove the theorem we begin by observing that if we can find in any way an n-ple of functions y1(x), y2(x), . . . , yn(x) satisfying the following system of n (non-linear) integral equations of the Volterra type⁸

(13)

such an n-ple constitutes a solution of the normal system (7) which satisfies the initial conditions (8); for on differentiating (13) with respect to x we obtain the ith equation of system (7), and we need only put x = x0 in (13) to verify that the function on the left-hand side assumes the value e9780486299297_i0027.jpg for x = x0.

A solution of (13) for which the functions y¡(x) may not be differentiable everywhere is called a solution of the differential system (7) in the sense of Carathéodory.

Now supposing that the index i takes the values 1, 2, 3, . . . , n, we construct an endless sequence of n-ples of functions by the recurrence formulæ

(14)

(Note that here the upper suffices do not denote derivatives.) We shall show first that given a positive number M such that throughout the entire domain D the continuous functions fi satisfy the inequalities

(15)

then for values of x in the interval (x0, x0 + δ) where δ is the smaller of the two numbers a and b/M (i.e. δ = min. (a, b/M)), the functions e9780486299297_i0030.jpg (x), e9780486299297_i0031.jpg (x), . . . , are always contained between e9780486299297_i0032.jpg –b and e9780486299297_i0033.jpg + b.

We have, recursively,

This is of importance because it ensures—provided that x remains in the interval (x0, x0 + δ)—that the functions fi having for argument y some one n-ple ( e9780486299297_i0035.jpg , e9780486299297_i0036.jpg , . . . , e9780486299297_i0037.jpg ) as defined above satisfy the hypotheses of the theorem, and, in particular, the Lipschitz conditions.

We now show that the successive n-ples defined above tend uniformly to an n-ple limit

(16)

i.e. given a positive number e as small as we please, we can always find an m0 such that, provided x lies between x0 and x0 + δ and i between 1 and n, we have for m > m0

(17)

For employing the first of the inequalities established above, viz.

and applying to the functions fi the Lipschitz inequality (11), we can write

where Ai is a suitable positive constant. Similarly

and so on. Thus, in general,

or

(18)

From this it follows that the series

(19)

is majorized by the convergent series of positive constants

and is therefore absolutely and uniformly convergent in the interval (x0, x0 + δ). Hence, since its sum to (m + 1) terms is e9780486299297_i0047.jpg (x), it follows that, for any x in (x0, x0 + δ) and for m > a suitable m0, i,

where Yi(x) denotes the sum of the series (19), i.e. the limit of e9780486299297_i0049.jpg (x) as m → ∞.

We need now only take m0 as the greatest of the numbers m0,1, m0, 2, ... , m0, n, to establish the result enunciated above.

The last step now to be done, consists in showing that the n-ple limit (16) satisfies the system of integral equations (13), i.e. that the relations

(20)

are identities.

To do this it is convenient to put

The recurrence formulæ (14) now yield

from which, by rearranging and subtracting from both sides of this equation

follows the identity

But in virtue of the Lipschitz conditions and (17) we have

and thus, by taking absolute values on both sides in the preceding identity, we deduce

In these inequalities the first term is some non-negative number, while the last term can be made as small as we please by suitably choosing ε; thus the first term must be equal to zero, i.e. (20) is satisfied identically.

Finally, we show that the n-ple (16) constitutes the unique solution of the system of integral equations(13).

For suppose the n-ple

(21)

is also a solution of system (13). Clearly the functions e9780486299297_i0058.jpg (x) must be continuous because so is the right-hand side of (13); also e9780486299297_i0059.jpg . Thus replacing (if necessary) the interval (x0, x0 + δ) by a more restricted interval (x0, x0 + δ′) we may suppose that

Now employing (15) and the equality

we deduce that

On the other hand, by subtracting term by term from the preceding equality the recursive formula (14) we obtain

from which, by taking absolute values on both sides and applying the Lipschitz inequality, it follows that

Thus, by putting m = 0 and using the penultimate inequality, we derive

whence, by putting m = 1 and using the result just obtained,

and so on. Thus in general we obtain

from which, since the general term in a convergent series must tend to zero, it follows that

i.e.

showing that the n-ple (21) is identical with the n-ple (16).

This completes the proof of the theorem of existence and uniqueness.

4. Additional remarks

It is repetitive to stress again the importance of the result just obtained, but it provides the rigorous basis of the whole theory of differential equations. It is not only the result itself which is important but also the method of successive approximations⁹ which establishes it, for this method may be extended in various ways; also it furnishes a process of calculation which, although generally laborious, nevertheless effectively produces as close an approximation as desired to the solution whose existence and uniqueness have just been established. We are therefore dealing with a constructive method of proof. Several examples which we shall presently discuss will illustrate this.

We can, in some cases, widen the scope of the result derived in the preceding section. First, the behaviour which is described to the right of the point x0 may be repeated to the left; thus if the hypotheses of the theorem are satisfied within the domain D′, in place of the domain D defined by the inequalities (12), where within D′ the previous limits on y remain valid while x varies in the interval (x0–a′, x0) where a′ > 0, then the theorem is valid in the interval (x0–δ′, x0) where δ′ is the smaller of the two positive numbers a′ and b/M. Hence if the hypotheses are actually satisfied for x0–a′ e9780486299297_img_8806.gif x e9780486299297_img_8806.gif x0 + a, the existence and uniqueness are established in a two-sided interval about x0, viz.

x0–δ′ e9780486299297_img_8806.gif x e9780486299297_img_8806.gif x0 + δ

A further important point to be noted is that the n functions Yi(x) constituting the solution described by the fundamental theorem may be considered as the sums of the uniformly convergent series (19). Thus in the case in which the functions fi are continuous functions of one or more parameters λ, µ, ... , e9780486299297_i0070.jpg (x) are evidently continuous functions of the same parameters; hence the functions Yi(x) must be continuous functions of λ, µ, . . . , as the sum of a uniformly convergent series of continuous functions is also a continuous function.

Therefore with suitable adjustments to the proof, as indicated above, the fundamental theorem may be stated as in the following paragraph, in which the initial values of the yi are now allowed to vary within suitable intervals and consequently in which the functions Yi(x) are continuous functions of the initial values e9780486299297_i0071.jpg . This turns out to be very useful.

The most general statement of the fundamental theorem is as follows:

Given a normal differential system of the form (7) and an n-ple of constants β1, β2, . . . , βn; if two positive constants a and b can be determined such that within the domain D defined by the inequalities

(22)

the functions f1, f2, . . . , fn are continuous and satisfy Lipschitz conditions in the variables y1, y2, . . . , yn within the given ranges, then a positive number δ may be found (δ e9780486299297_img_8806.gif a) such that, given any set e9780486299297_i0073.jpg , e9780486299297_i0074.jpg , e9780486299297_i0075.jpg , . . . , e9780486299297_i0076.jpg of initial values satisfying

(23)

the differential system (7) possesses one and only one solution which satisfies the initial conditions (8) in the interval (x0, x0 + δ).

If in the domain D the functions fi are all in absolute value less than M, δ may be taken equal to the minimum of a and b/(4M).

This more general form of the fundamental theorem can be established by suitably amending the proof given for the weaker theorem. See, for example, Sansone (47), chapter I. This reference may be consulted also in regard to the following problem which we can merely mention here.

What happens if the functions fi are continuous but fail to satisfy the Lipschitz conditions in the variables yh within the given ranges?

It is fairly evident that even under these more general hypotheses the existence part of the theorem remains valid, but in general the uniqueness part is no longer true, as will be shown in an example we shall consider at the beginning of the following chapter.

In view of the character of this book we cannot consider in detail this important extension of the existence theorem¹⁰—due to Peano—which requires less simple expansions than we have used, and which in fact belongs naturally to a field that we cannot even touch upon. This extension is concerned with existence and uniqueness theorems in the large, i.e. with theorems of this kind valid not only in a sufficiently small region about a certain point, but valid throughout a whole domain in which the equation satisfies certain conditions.

We note finally, in connection with these last points, that the solution whose existence is established by the fundamental theorem in the interval (x0, x0 + δ) may, in general, be extended into a further interval (x0 + δ, x0 + δ + δ1) by applying the same theorem to the point x1 = x0 + δ with initial values equal to the final values of the solution obtained immediately previously, and so on. It must not however be assumed that this process can be continued until the conditions on the functions fi are no longer satisfied, i.e. it must not be assumed that if the set of numbers x0 + δ, x0 + δ + δ1, x0 + δ + δ1 + δ2, . . . , has finite upper bound ξ the point x = ξ is necessarily a point in which at least one of the functions fi ceases to satisfy (at least for some set of values of the yi’s) the conditions demanded in the fundamental theorem.¹¹

To illustrate this point, it is sufficient to consider the simple example of the single (non-linear) equation of the first order

(24)

whose general integral is

Clearly the integral

(25)

which for x = x0 takes the value y0, cannot be continued beyond the point x = x0–1/y0 (where y becomes infinite) despite the fact that at that point there is no singularity of (24)—in which equation, in fact, x does not appear explicitly.

5. Circular functions

It is instructive from the point of view of the general theory, to study the differential system

(26)

together with the initial conditions

(27)

Disregarding meantime the familiar properties of its solutions which are y1(x) = sin x, y2(x) = cos x, we try to read off from equations (26) and (27) the principal properties of the solutions of the system.

We first investigate how the method of successive approximations used in the proof of the fundamental theorem applies in this case. As in this example,

the recurrence formulæ for the successive approximations take the form

and elementary calculations yield

and so on.

Hence successive approximations give successive partial sums of the series (but here with each sum repeated twice over)

(28)

which are everywhere convergent and which are the expansions as power series of the functions sin x and cos x. In other words, in this actual case, the method of successive approximations yields the power-series expansions of the two solution functions, and as these power series are convergent for all x we deduce that the two solutions in question are bounded.

Several properties—for example, that sin x and cos x lie between–1 and +1, or that they are periodic of period 2π—may be deduced¹² fairly easily from the series (28); but it is of interest, as has already been suggested, to look for a method of reading off these properties directly from the differential equations.

We observe initially that by multiplying the first of equations (26) by 2y1, the second by 2y2 and adding, we obtain

But the left-hand side is the derivative of e9780486299297_i0088.jpg + e9780486299297_i0089.jpg , and therefore

On account of the initial conditions (27) this becomes

(29)

From equation (29) follow several corollaries. First,

|y1| e9780486299297_img_8806.gif 1, |y2| e9780486299297_img_8806.gif 1

Second, y1 and y2 can never vanish simultaneously; also their zeros are all simple, for if y1 and y′ 1 vanish simultaneously¹³ at a certain point (as happens at every multiple zero of y1) then, by the first equation of (26), y2 = 0 at that point.

But do the functions y1 and y2 actually vanish at any points?

For the function y1(x) this is evidently so, since the series which appears in the first equation of (28) vanishes at least for x = 0. For the function y2(x) there may in fact be dubiety, but this is easily removed by a reductio ad absurdum argument as follows.

If y2 never vanishes we may suppose y2 always positive for x > 0 since y2(0) = 1; then y1(0) = 0 and y′1 = y2 > 0; thus the function y1 is always positive for x > 0, so that for x e9780486299297_img_8807.gif ε > 0, y1(x) e9780486299297_img_8807.gif η where y1(ε) = η. But since y′2 =–y1, the preceding inequality obviously implies

which is incompatible with the fact that y2(x) is always bounded—and the result follows.

The differential system (26), and in fact all systems in which the independent variable x does not