In fact, in this case the differential formula of § 198 reduces to that of § 195, and is the first of the two examples of the result of § 195 given in that paragraph. 200. X=xƒ(X'). A differential formula only slightly different from that of § 198, and to be dealt with in the same general manner, is log (Ca)=4(X') X' is to be eliminated algebraically so as to leave the integral equation between x and X. 201. X=nxX'. A particular case of the last formula, which is also at the same time a particular case of § 196, is whence by direct integration of each side 202. XxX' +ƒ(X'). 1 X=Cx If the differential equation be of the more involved form X=xX' +f(X'), by taking the x-gradients on both sides, there is obtained, X' cancelling out from the two sides, since X=xX' +ƒ(X') and X'=С1. This is a partial integration of the given differential equation. The other solution is From this and x+ƒ'(X') = 0 | X=xX' +f(X') treated as simultaneous equations, X' may be algebraically eliminated, leaving an equation giving X in terms of x either explicitly or implicitly. Let this equation be symbolised by p(x,X)=0. This p(x,X)=0 is a second partial solution of the given differential equation. The combination of these two partial solutions gives the complete solution in its most general form, which, in application to whatever physical problem may be in hand, must be particularised by the insertion of the "limiting conditions." These limiting conditions sometimes exclude one of the "partial" solutions as impossible, leaving the other partial solution as the full true solution of the particular physical problem in hand. The solution of a more generalised form of this differential equation is given in § 210, the method of solution depending on that of § 208. A form differing from the last only in the sign of xX' is X+xX' =ƒ(X'). Here X+xX' is the x-gradient of xX. i THE CALCULUS FOR ENGINEERS. Therefore, the integration gives and, if X" be expressible in terms of X' alone and the function ƒ(X') X be integrable by dX', this integration will give an equation between x, X, and X', between which and the original equation, X' may be algebraically eliminated, leaving one involving only x and X. 203. Homogeneous Rational Functions. If the relation between x, X, and X' be found in the form (axTM + bâm¬1X+ câm¬2X2 + ----)X′ = Axm + Bæm-1X + Câm−2X2 +---where the (x,X) functions on the two sides are both "homogeneous" of the mth degree, that is, where each consists of a series of products of powers of x and X, the sum of the two indices in each term being m; then by dividing each side by 2m, this may be converted into an equation in X = X X or X=x; therefore X'+xx'. Dividing by am, the differential equation becomes (a+b+cx2+----) (+x*') = A+B+C¥2+-- From this is easily deduced dx = a+b+c+------ x A+ (B-a) + (C − b) ¥2 + · .dx. The integral of the left side is log. Therefore, if that on the right is directly integrable* to a function of X, say 6(X)=6( then the equation gives the desired integral relation between x and X. A convenient shorthand symbol for such a homogeneous (x, X) function of the mth degree is f(xm-r, X"). The two such functions * See Classified List, III. A. 14. may be called fam-r, X") and F(xm-r, X"). The given differential equation may then be written X'ƒ(xTM-r, X') = F(1⁄2”—”, X”). Dividing f(xm-, X") by xm we obtain the same function of 1 and as f(x, X") is of x and X. The quotient may, therefore, be written f(1-,), and similarly that of F(am-r, X") by x may be written F(1m-, *). The integral equation then appears as 204. Homogeneous Rational Functions. The last form is a particular case of a more general one involving the first power of X' only, and the ratio only of X to x. Call this ratio as in last article, so that as before X' = +x'. The present more general form of differential equation may be written 205. X=xf(X'). A form of differential equation of cognate character is that solved in § 200, namely, One solution of this is given in § 200. Otherwise, it may possibly be more easily solved algebraically so as to give X' explicitly in THE CALCULUS FOR ENGINEERS. terms of . Let f-1) denote the inverse of the function f(). Then this algebraic solution would appear as X' =ƒ ̃1(*) of which, according to last article, the integral solution is 206. X' (Ax+BX+C) ÷ (ax + bx + c). = A differential equation bearing a resemblance to that of § 203 is (ax+bX+c)X' = Ax+BX+C. If the two constants c and C did not appear, then by dividing by x, each (x,X) function would be converted into one involving the ratio only of the two variables. shifting parallelly the axes of co-ordinates from which x and X are But c and C can be got rid of by measured, which change does not affect X'. new co-ordinates, then it is easily shown that the axes must be If x and X be the shifted so as to make Then, since dx dx = Bc-bC dx dx X', by dividing out by x, there results an equation of the form dealt with in § 203. 207. Particular case, B = a. If in the equation of last article B = with the common factor a combine to make the complete increment -a, then the two terms of xX. Thus the equation then reduces to (bX+c)dX+a(Xdx + xdX) = (Ax + C)dx the integration of which gives ¿bx2+cX+axX – †Ax2 - Сx + K −0 in which K stands for the integration constant. 208. X'+XX' = E. = Let and E be given functions of x, of which is a function whose integral by dx can be directly found, namely. Then, if the differential equation between X, X', and x be found to be X' + XX' =E; |