and is therefore of the form which has already been discussed. A In the final result we shall have to replace y by y-=; -; and An a, are the n roots of the characteristic of An Also from (120) the same result will be derived. Let x = A; Also if the characteristic has impossible roots, or has equal roots, the results are similar in form to those investigated 458.] The investigations of the preceding articles will have shewn that the solution of a linear differential equation with a second member depends in a great measure on the solution of it without the second member; for if the general integral of the latter can be determined, that of the former will also be determined if a certain given function of a single variable can be successively integrated, and added to it. A method for this purpose, other than the preceding, was devised by Lagrange, and is commonly called Lagrange's method of variation of parameters. This I proceed to explain. Take the two following linear differential equations expressed in terms of derived functions: y(n) +A1y(n−1) + A¿Y(n−2) + + An-1Y+AY = X, ... (128) 2(") + A, 2(n−1) + A2≈(n-2) + ... + A„-12′ + A„≈ = 0; (129) and suppose Z1, Z2, ... Z, to be n particular integrals of (129), so n then it is always possible to determine n functions of x, viz., u1, u2,... un, so that the general integral of (128) may be that is, the integrals of (128) and (129) are of the same form ; but the arbitrary quantities C1, C2, ... C,, which are constant in the integral of (129), are functions of x in that of (128). To exemplify the process before I enter on the general theory, I will take first the case of the linear differential equation of the first order, viz. dy where the coefficient of y and also the right-hand member are functions of x. Now omitting the second member we have where c is an arbitrary constant. Now Lagrange in the method of variation of parameters supposes the integral of the equation with its second member to have the same form as that of the equation without the second member; the constant of integration in the latter case however being replaced in the former case by a determinate function of x; thus he takes (134) to be the integral of (133), when c is a function of x. To determine c, let us differentiate (134), and substitute in (133). From (134) we have e−ff(x)dxc+ y = e-fixa{c + fr(x) effinde de which is the same solution as that given in (69), Art. 382. The method is called variation of parameters, because c, which is a constant parameter in (134), is made to be that function of a variable which is given in (136). Let us now apply the same theory to the general linear differential equation (128). Let us suppose (132) to be its integral; then differentiating, dy dx dz du =Σ.U + 2.2. ; (137) dx Differentiate again (137) subject to this condition and we : and continuing the same process, and making similar substitu we have and substituting these values throughout in (128), we have d" z dn-1 z du z.U {z(") + A2 z("−1) + ... + An-12′+A„~} +2. dn-1 z du = X. Now of the expression on the left-hand side of this equation, 21, the first part vanishes, because z1, Z2, ... Z, are particular integrals of (129); and therefore Hence (131) is the general integral of (128), the values of the u's being found from the following system of equations : These equations will of course in general give n different values of u, u,... u,' in terms of the z's and of x, and each value will have x as a factor; suppose the other factors to be V1, V2, ... Vn: then + [v2 x dx ; (139) and substituting these in (131), we have the general integral of (128). In this process we have made no restriction as to the coefficients of the given differential equation; they may be either constants or functions of x: if however they are constants, and are the roots of the characteristic of (129), and these must be substituted in the series (138); and thence may be deduced the values of v1, v,... v,, which are required for (139). The following are examples of this process. and substituting in the given differential equation, we have from which, combined with the supposition made above, we have of which the integral by the processes of the preceding Articles is where c1 and c2 are arbitrary constants of integration. : then Let these values be substituted in the given differential equa so that the integral of the given differential equation is (141) |