A Treatise on Differential Equations

CHAPTER IX.

ON DIFFERENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST.

1. THE typical form of a differential equation of the nth order is given in Chap. I. Art. 2. We may, by solving it algebraically with respect to its highest differential coefficient, present it in the form

dry
dx = f (x, y, day, day, dr.)(1).

dx'

dx"

Its genesis from a complete primitive involving n arbitrary constants has been explained, Chap. 1. Art. 8.

Conversely, the existence of a differential equation of the above type implies the existence of a primitive involving n arbitrary constants and no more; and a primitive possessing this character is termed complete..

The converse proposition above stated, is one to which various and distinct modes of consideration point, but concerning the rigid proof of which opinion has differed. The view which appears the simplest is the following. If, as in Chap. II. Art. 2, we represent by Ap (x) the increment which the function(x) receives when receives the fixed increment Ax, and if we go on to represent by A' (x) the increment which the function Ap (x) receives when x again receives the same fixed increment Ax, and so on, then it is evident that the values of Ap (x), A3p (x), &c., are fully determinable if the successive values of the function (x) in its successive states of increase are known. Thus since

A$ (x) = $ (x + Ax) − $ (x),

we have by definition

▲3p (x) = ▲ {$ (x + ▲x) − p (x)}

· {☀ (x + 2Ax) − 4 (x + ▲x}} − {† (x + Ax) − $(x)}

4(x+2Ax) – 24 (x + Ax) + P(x),

and so on. Conversely if

(x), Ap(x), A2p (x), &c.

are given, the successive values of the function (x), viz. the values (x+Ax), 4(x+2Ax), &c., are thereby made determinate. Geometrically we may represent (x) by y, the ordinate of a curve, or of a series of points in the plane x, y, and therefore functionally connected with the abscissa x.

Now the view to which reference has been made is that which, 1st, presents the differential equation (1) as the limiting form of the relation expressed by the equation

Ax-f(x, y, Ay, Ay, Ay)...

Δα

Ax-1

.(2),

Ax approaching to 0; 2ndly, constructs the latter equation in geometry (the arithmetical or purely quantitative construction being therein implied) by a series of points on a plane, of which the first n, viz. those which answer to the co-ordinates x, x + Ax, ... x + (n-1) Ax, have the corresponding values of y arbitrary, while for all the rest the values of y are determined; 3rdly, represents the solution of the differential equation as the curve which the above series of points in their limiting state tend to form. According to this view, the n arbitrary points in the constructed solution of the equation of differences (2) give rise to one arbitrary point in the limiting curve, accompanied by n-1 arbitrary values for the first n-1 differential coefficients of its ordinate. And this mode of consideration appears the simplest, because it assumes no more than the definition itself demands of us when we attempt to realize the geometrical meaning of a differential coefficient as a limit. We may however add that when by the consideration of the limit, the mere existence of a primitive has been established, other considerations would suffice to shew that in its complete form it will involve n arbitrary constants and no more. The fact that each integration introduces a single constant is a direct indication of the fact. An indirect proof of a more formal character will be found in a memoir by Professor De Morgan (Transactions of the Cambridge Philosophical Society, Vol. IX. Pt. II.).

The above theory may be illustrated by the form in which Taylor's Theorem enables us to present the solution of a differential equation of the nth order, as will be seen in the following Article.

Solution by development in a series.

2. Reducing the proposed equation to the form

and differentiating with respect to x, the first member becomes

while the second member will in general involve all

d"y
dx"

If for the last we

the differential coefficients of y up to substitute its value given in (3), the equation will assume the

form

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Thus is expressible in the same manner as

dxn+1

dx"

terms of x, y, and the first n 1 differential coefficients of y.

Differentiating (4) and again reducing the second member by means of (3) we have a result of the form

and in this form and by the same method all succeeding differential coefficients may be expressed.

Hence reasoning as in Chap. II. Art. 12, we see that supposing y to be developed in a series of ascending powers of where xo is an assumed arbitrary value of x, the coefficients of the higher powers of x-x, beginning with (x-x)" will have a determinate connexion, established by means of the differential equation, with the coefficients of the inferior powers of x-x. The latter coefficients, n in number,

beginning with the constant term which corresponds to the

index 0, and ending with

d"-ly

which is the

1.2.3...(n-1) dx"-1 coefficient of (x − x)" ̄1, will be perfectly arbitrary in value.

To exhibit the actual form of the development let Yo

Yn be the arbitrary values assigned to y,

N-1

dix'

...

dx"-1

when

x=x Also let ff, f, &c. represent the values which the second members of the series of equations (3), (4), (5) assume when we make in them x=

x; then

y = Y。 + Y1 ( x − x.) + 1/2 1⁄2 (x − x )3... +

(x-x)+1...&c. ad inf..........(6).

In this expression the arbitrary values of y and its n— first differential coefficients corresponding to an assumed and definite value of x, viz. yo, y1, ... Y are the n arbitrary constants of the solution, the values of fn, fn+1, &c., being determinate functions of these, and therefore not involving any arbitrary element.

Any function of arbitrary constants is itself an arbitrary constant, and thus it may be that an equation has effectively a smaller number of arbitrary constants than it appears to have from the mere enumeration of its symbols. As a general principle we may affirm, that the number of effective arbitrary constants in the solution of a differential equation while on the one hand equal to the index of the order of the equation, is on the other hand to be measured by the number of conditions which they enable us to satisfy. Systems of conditions to be thus satisfied will indeed vary in form, but there is one system which we may consider as normal and to which all other systems are in fact reducible. It is that which is described above, and which demands that to a given value of x a given set of simultaneous values of y and of its differential coefficients up to an order less by 1 than the order of the equation shall correspond. Conversely, the arbitrary constants

of a solution may be said to be normal, when they actually represent a simultaneous system of values of y and its successive differential coefficients up to the number required.

Ex. Given

d'y
dx

+ y2. Required an expression for y

in the form of a series such that when x = 0,

assume the respective values of c and c'.

Differentiating, we have