Vn) dy * = •• • A (*» Vv Vv • • •» Vn) (46) fn(x,yvyv->yn) On multiplying all the denominators by the same factor, we get a function of variables x, yv y2, . . , yn instead of unity in the denomi nator of the first fraction. If we denote the variables as cCj, x2, . . f xm xn+1 for the sake of symmetry, the system of differential equations (42) can be written in the form dxl Xx dx2 X2 drrn Xn d# n +i Xn+1 .
13 We have a parabola for ( 7 ^ 0 , and the axis OX for (7 = 0. Equations (82) become: y = C (x - C)2; (x -C)(x3C) = 0. The second equation gives G = x or O = ic/3. Substitution in the first equation gives us either y — 0 or y = Ax3/2 7. The first curve y = 0 is axis OX, which belongs to the given family of curves; whereas the cubical parabola y = 4#3/27 is the envelope of the family. 5. We take the chords of the circle of unit radius, centre at the origin, that are perpendicular to OX and we draw fresh circles with the chords as diameters, thus obtaining a family of circles.
1453. 1311**, which gives a good degree of accuracy for t near zero. 14. Graphical methods of integrating second order differential equations. There is a corresponding curve for every solution of a differential equation of the nth order, and, as in the case of first order equations, we shall call the curve an integral curve of the equation. In the case of a first order differential equation, there was a corresponding tangent field . We now explain the geometrical significance of the second order equation y" = f(v,y,y')- (12) f It is to be noted that we obtain the series for x'x and xJ, not by differentiating the series for xlf but by applying Taylor's formula to x'x and x": .
A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.)