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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 helppo + 4 + Bar and + ... + Po-hoa + PnY = 0; (64) and let the n particular integrals be nu, 92... Mm.

Substitute in (64) for y, y = n/ ,dx ; then we have du-lv, , dn, dn–202, n(n-1) dan, d"-3y, dan lae

n

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+{1,0 + en Jo, des}

+ Patav, dx = 0. (65) Now the coefficient of /v, dx = 0; consequently, dividing through by na, we have

Let

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= Qy, and let the coefficients of the succeeding terms be Q3, Q3, ... Qn-1: then (66) becomes da-lu,

22 dark-3

-3 + ... + Qn-14, = 0. (67) Now of this equation the (n-1) particular integrals are d ng d ng d nm.

(68) dx ne dx ñ dx in 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 Ry, Rag, ... Rr-2, we shall have

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and by continuing a similar process in the equation which involves Ry, Ry, ..., 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 P, expressed wholly in terms of the n'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 F, (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

ni = 041%, 12 = pagit, n3 = (434. Let the equation be

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

v dx vanishes, we have

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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. 11.

4 M

10.) To investigation of the providing metion show that the

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
take to be
daydn-lydn-

sy

d y.. inn +41 Imn=1 + 1, Ton- + ... + An-lain + 4y = x, (79) where Ay, A2, ... A, are constants, and x is a function of x.

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

y(n) +47 y(n-1) + A2 y(n-2) + ... + An-1% + Any = x; (80) and introducing certain undetermined constants 0, 0%, 0%", ... o(n-1), we may put (80) in the form

{y(n-1) +6°y(n–2) +0% y(n-3) + ... +6(0–1)y} +(A-8)g(*-1)+(^-8x(x-2)+...+(A-1-8(+-2)+ Ang=x; (81) and let us make the following substitutions;

y(n-1)+o'y(n-2) + ... +6(n-1)y = 1;
4,-0= 0; 12-0'= 00', A3-0"= 00......,?

(83)
An-1-A(n-1) = 00(1-2), An= 00(n-1);
so that (81) becomes din

(84)

(82)

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and for 0 let - a be substituted: then from (83) we have

a" + A, an–1 + A, an–2 + ... + An-1 a+A, = f(a) = 0; (86) 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 aj, ag, ... ani then there are n different values of (85), viz.

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(93)

= I. Am Am 1m +.4mx;

(94) 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 a.dm x = 0, and therefore when 2.dm = 0, we have

y'= 2. Am Am Nm; and therefore after differentiation y'= .amIm nm + 8.Am Am X;

(95) and as y" must also be of the same form as y, .Amdm X = 0;

... y'= x.am? Am Nm;

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