An elementary course of infinitesimal calculus . This represents a system of rectangular hyperbolas whose axescoincide in direction with the asymptotes of the former system. Ex. 2. To find the curves orthogonal to the circles a? + f + ¥ = Q (8), where /a is the variable , we have xdx + {y + ij) dy = 0,and therefore, for the trajectory, xdy - {y + It) dx = /x, between this and (8), we find 2xy^ + {a?-y-l^) = 0 (9), or x^^-y^-o^ + k (10). This is linear, with y^ as the independent variable. The in-tegrating factor, as found by the rule of Art. 177, or
An elementary course of infinitesimal calculus . This represents a system of rectangular hyperbolas whose axescoincide in direction with the asymptotes of the former system. Ex. 2. To find the curves orthogonal to the circles a? + f + ¥ = Q (8), where /a is the variable , we have xdx + {y + ij) dy = 0,and therefore, for the trajectory, xdy - {y + It) dx = /x, between this and (8), we find 2xy^ + {a?-y-l^) = 0 (9), or x^^-y^-o^ + k (10). This is linear, with y^ as the independent variable. The in-tegrating factor, as found by the rule of Art. 177, or byinspection, is l/ar*. Introducing this we have dx\xj ^s>? 480 INFINITESIMAL CALCULUS. [CH. XI whence or ^ = -x + 2X, X X (11). X being arbitrary. The original equation represents a system of coaxial circles,cutting the axis of x in the points {±k, 0). The trajectories (11)consist of a second system of coaxial circles having these pointsas limiting points; viz. if we put X = ±k we get the point-circles {x + kf + f = 0 (12); see Fig. 148,. Fig. 148. If the equation of the given family of curves be in polarcoordinates, thus f(r,e,G) = 0 (13), 178] DIFFERENTIAL EQUATIONS Of FIRST ORDER. 481 and if ^, ^ denote the angles which the tangents to theoriginal curve and to the trajectory make with the radiusvector, we have in like manner tan <f) = — cot 0. Hence the differential equation of one system is obtainedfrom that of the other by writing \dr , rdOrdO dr Or, differentiating (13) we have %dr + l%rde = 0 (14), and therefore, for the trajectory, ^lrdd--%dr = 0 (15). dr rod The elimination of C between (13) and (15) leads to thedifferential equation of the required system. Ex. 3. In the circles r = c cos 5 (16)) which pass through the origin, and have their centres on theinitial line, we have - = -ta,ned9 (17), r and therefore, for the trajectory, rd$ = ta,nedr, or —^ootOdd (18). Integrating, we find log r = log sin 6 + const., or r = csin^ (19), which represents another
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