449.] THEOREM VI.—If n particular integrals of a differential equation, which is without the second member, are known, the coefficients of the several terms are functions of these integrals, and may be found by a process analogous to that of forming an algebraical equation whose roots are given.

Let the differential equation be of the nth order, and of the form

and let the n particular integrals be 71, 72, ... nn.

Substitute in (64) for y, y = N1

Sv1da; then we have

dx

+P_{10, + d /o,dr}
+ Panft, dr = 0. (65)

Now the coefficient of v1dr = 0; consequently, dividing

n dni

Let

ni dx

+P1 = Q1, and let the coefficients of the succeeding

terms be Q2, Q3, ... Qn-1: then (66) becomes

Now of this equation the (n−1) particular integrals are

let us therefore repeat in (67) the same process as that to which

(64) has been subjected; then if the successive coefficients of the transformed equation, which will be of the (n-2)th order, are R1, R2, ... R-2, we shall have

and by continuing a similar process in the equation which involves R1, R,..., we shall find an equation whose order is the (n-3)th, and shall be able to express P, in terms of others of the original particular integrals: and so on, until finally we arrive at a value of P1 expressed wholly in terms of the ŋ's.

By a process exactly similar, the other coefficients of (64) may be found in terms of the particular integrals. And thus in general, if F1 (x), F2(x),... F ̧(x) are n functions of x, and it is required to determine a linear differential equation of which these are n particular integrals, we can determine the coefficients of it in terms of the particular integrals. This case is plainly analogous to that of the formation of an algebraical equation of which the roots are given.

In illustration of this process let it be required to form the differential equation of the third order, of which three particular integrals are

then, as ea is a particular integral of (70), the coefficient of

[v,da vanishes, and the transformed equation is, after division

by ea1*,

of which two particular integrals are, by reason of (53),

(71)

then substituting in (71), and observing that the coefficient of

vada vanishes, we have

dva

dx

+(P2+α1+2α2) v2 = 0;

(74)

of which, by reason of equation (44), e(a1⁄2-ɑ2)x is a particular integral; therefore substituting we have

d2v1

dx2

dv1 dx

(75)

+(2α1—α2-α3) +(a ̧2—2α ̧a‚—2а1αz+P2) v1 = 0; (76)

and of this e(3-1) is a particular integral: therefore substituting,

(77) and substituting in (70) for P, and P2, and noticing that ea is a particular integral of (70), we have after substitution

And this equation might also have been found as follows: Since ea, e, ea are particular integrals, we might substi

tute these in it, and thereby obtain these equations,

of which three cubic equations a1, a, a, are evidently the roots; therefore

Similarly it may be shewn that the equation, of which particular integrals are x-1 and x2, is

SECTION 3.-Integration of Linear Differential Equations of the nth Order, whose Coefficients are Constants, with or without Second Members.

450.] The investigations of the preceding section shew that the PRICE, VOL. II.

4 M

integration of an equation of the linear form with the second member depends on that of the same equation without the second member, and on a multiple integral the element-function of which involves the second member: in the present and the future sections therefore we shall, if it is convenient, consider properties of linear differential equations, without the second members, and the reader will observe that the generality of the investigation is not diminished thereby. There are many processes of solution, which will be considered consecutively. The general type I shall

FIRST METHOD. Let (79) be expressed by means of Lagrange's notation of derived functions; then we have

y(") + A1y("−1) + Ag y("~2) + ... + An-1Y+Any = x; (80) and introducing certain undetermined constants e, 0′′, 0",... (-1), we may put (80) in the form

d

+0′′

dx {y("− 1) + 0′ y("− 2) + 0′′ y(n−3) + ... + 0("−1) y}

+ (4,−0′ ) y (n−1) + (^2—0′ ) y(n−2) + ... + (A„—1—0(”−1)) y′+ A„y=x; {(8])

and let us make the following substitutions;

(86)

and for 0 let

-a be substituted: then from (83) we have

An-1a+An

a” +▲1 a”−1+▲2 a®−2 + n-1 n-2 + ... + An-1 a +11 = ƒ(a) = 0; the resemblance of which to (79) in its powers and its coefficients is evident; and as we shall hereafter refer to this equation, it is convenient for it to bear a particular name: let it therefore according to a received nomenclature be called the characteristic equation of (79).

Now suppose the n roots of this equation to be unequal and to be a1, a,... a,; then there are n different values of (85), viz.

er{fewxdx+9}; ex{fe~~*xdx+Q};... (87)

which may be denoted by 71, 72, 7; also let the values of o', 0", (n-1) corresponding to these roots be 01, 0",... 01(n−1), 02, 02 02("−1),... 0„′, 0",... 0,("-1); then from (82) we have

"

29

...

the following series :

n

y(n−1)+01'y(n−2) +01⁄2′′y(n−3) + ... +0 ̧(n−1) y

n

Ξημ

y(n−1)+01⁄2'y(n−2) +0,1⁄2′′y(n−3) + +02(1)y = n2i

...

y(n−1)+0„'y(n−2) + 0„′′y(n−3) + ... + 0, ("-1) y =

(88)

(89)

the values of y', y′′,... y(n−2), y(n-1) are evidently similar in form. Now the value of y given in (89) is of the form

where A1, A2, A, are constants and functions of the 's; and these are assigned by (89); but it is easier to discover them by the following method.

451.] Let us employ a concise notation and represent (90) thus ;

where x indicates the sum of a series of terms found by giving successive values to m from 1 to n inclusively; then

and observing the remark made in the sentence following equation (89), that y' must be of the same form as y, and as this can be the case only when .λm X = O, and therefore when 0, we have

and as y" must also be of the same form as y, z. ɑmλm x = 0;

(95)

.:._y"= x.am2 λm Im ;