101.] Differential equations also sometimes arise in another way it is often more convenient to express the relation which holds between two given quantities or functions by a differential than by its primitive equation: the differential equation indeed, as will be seen hereafter, can frequently be found, when the primitive is beyond our powers of expression. Here however I shall give only one or two examples wherein differential equations are formed.

Ex. 1. It is required to express in the form of a differential equation the relation between cos nx and cos x.

which is a differential equation of the second order. If we assume un to be expanded in a series of ascending powers of z, the coefficients of which are to be determined, this equation will enable us to determine the coefficients, because it is to be satisfied by them for all values of n.

In this equation it will be observed that cos x is the equicrescent variable.

Ex. 2. It is required to express in the form of a differential equation the relation which exists between cos x and the series

un = cos nx + cos (n − 1) x + + cos 2x + cos x.

...

The sum of the series is found without difficulty; and we have

Let cos x = 2; .. dz= sin x da; let z or cos x be equicrescent; then differentiating twice, we have

which is the required differential equation of the second order. These two examples are sufficient to illustrate the process.

102.] Thus far the primitive equations have contained only two variables, and the differentials and derived-functions have been total: it remains still to exhibit the analogous processes, when three or more variables are implicitly or explicitly involved, and when the differentials are therefore partial. I shall consider two cases of this process. In the former the variables will be contained in an equation which is free from any undetermined or arbitrary function. In the latter I shall suppose one or more arbitrary functions to be involved, of which the subject variables enter in certain given relations, such as we have supposed and exemplified in Art. 52 and 53; and the derived functions of which will accordingly bear certain relations to each other dependent on the form in which the subject variables are combined, and will thereby admit of elimination.

Ex. 1. It is required to eliminate A, B, C and D from the equation Ax+By+cz+D= 0, and to express the resulting relation in the form of a partial differential equation.

AX+BY+CZ+D = 0.

Let us assume z to be the dependent variable, and take successively the x- and y-partial differentials; then, observing the

dx dy dy2

either one of which differential equations satisfies the equation. Of these results we shall hereafter see the geometrical meaning. Ex. 2. Determine the partial differential equations which arise in the elimination of a, b, c from the equation

Taking z to be the dependent variable, we have

and of the former of these two, taking the x-partial differential,

from which and (172) eliminating a2 and c2, we have

(174)

(175)

Similarly, if we take the y-differential of (173), and eliminate

Either (175) or (176) is the required differential equation.

Ex. 3. It is required to express a property of the following equation in terms of partial derived-functions, and independently

As many examples of the formation of these and similar differential equations will occur in the sequel, it is unnecessary to insert others here.

103.] Again, let us consider the formation of partial differential equations as they arise from the elimination of arbitrary functions, when the subject-variables of these functions are combined in particular relations. And first let us take the most simple case; that viz. of an explicit function of two independent variables, of the form

in which the form of F is undetermined, and wherein a and y enter in particular combinations, such as a divided by y, or x multiplied into y, and so on: so that the x- and the y-partial derived-functions of u will contain the same arbitrary functional term. Let us also suppose that (177) contains only one arbitrary function. Then the partial derived-functions having been calculated, we shall have two equations containing the same arbitrary function, which by means of them may be eliminated; and an equation will result in terms of (d), (d), the vari

2

ables and constants, some or all, of the primitive equation, and it will be independent of the arbitrary function.

If the primitive contains many arbitrary functions of x and y, with their subject variables combined in given relations, successive partial derived-functions may be formed, and thence equations will be obtained independent of the arbitrary functions, and in terms of the partial derived-functions. The process has already been employed in Art. 52 and 53 of Chapter II, and (56), (59), (62), (65), (76), (86) of those Articles are cases of it. The following examples are in further illustration.

Ex. 1.

u = ax + by + cf(mx+ny);

then if, as explained in Art. 53, the relation between ƒ and ƒ' is such that d.f(z) = ƒ'(z) dz,

let' be related to F so that d.F(z) = F(z) dz, then

let ƒ and ƒ' be related so that d.f(z) = f'(z) dz; then

We will now take a case wherein two arbitrary functions are involved, and two successive differentiations are required.

Ex. 4.

u = f(ax+by)+4(bx−ay) ;

then using a notation the same as heretofore,

... a

(du)

du

(d)

dy

du

dx

= a f'(ax+by)+b '(bx—ay),

= bf'(ax+by)—a d'(bx—ay);

+ b (du) = (a2 + b2) ƒ' (ax + by).

dy

And taking again successively the x- and the y-differentials of

d2u

= a (a2+b2)ƒ" (ax + by),

(dd) + b (d2u) = b (a2 + b2) ƒ” (ax + by) ;

dy dx

.. ab { (du) – (džu) } + (b2 —a2) (dx dy)

= 0.