All equations of the forma y = f(y) may be integrated by this process; for multiplying both sides by 2 dy, 2 dielny = 2 dyf(y); i rayon = 2 [f(y)dy +c; and of this the root is to be extracted, and a subsequent integration is to be effected. Ex. 8. Determine the curve whose curvature is constant. log (p+a) = log contentc; but as ay is a transcendental function of x, the next integration cannot be effected. Ex. 6. Determine the curve of which the radius of curvature is proportional to the normal. dy2, i 1 + diet = ()*; where k may be either positive or negative; dx = {(\*-1}dy. (1) Let k = 1; that is, the radius of curvature is equal to the normal; .. '= versin-124-(cy—yo)+; the equation of a cycloid, whose starting point is the origin, and whose base is the axis of x. Ex. 7. Determine the curve whose radius of curvature varies inversely as the abscissa. an equation which does not admit of further integration, but which represents the elastic curve. Also see Art. 329. Ex, 8. Determine the equation of the curve of which the radius of curvature varies as the cube of the normal. dy? \ * a’ya + k* (x -c)2 = at; the equation of an ellipse. The curve is a hyperbola if kz is replaced by – k; and is a parabola if no constant is introduced at the first integration. Ex. 9. Find the equation to the curve in which where ds is the length-element of the curve. Ex. 10. A form of differential equation which frequently occurs in subsequent investigations is any + k2y = 0. Let both terms be multiplied by dy; then integrating, due + k? (y2–62) = 0; and according as the upper or the lower sign is taken, we have y = b cos k (x -a), 464.] Next, let us consider homogeneous equations of the second order : the principle of homogeneity being estimated in the following manner; the variables x, y, and their differentials dx, dy, dły are considered to be factors of the first degree; and each term of the equation is of the same degree in respect of them; thus the equation, x3day-(y dx - x dy)2 = 0, is homogeneous and of the fourth degree. Now in such an equation let the following substitutions be made ; viz. y = xz; ii dy = x d2 +z dx ; (151) also let (152) and it is manifest that x will enter in the same power into all the terms, and therefore may be divided out; this property in fact is the characteristic of the equation; and thus the resulting equation will contain 2, v and Cy; for convenience of notation, p: so that from (151), and v may be expressed in terms of z and p by means of the given equation, and therefore by the last two members of the equality we shall have a differential equation of the first order in terms of p and z, whereby p may be expressed in terms of z: and therefore from the first two members of (153) we shall obtain a differential equation of the first order in terms of x and z; and this after resubstitution will give the required integral. Ex. 1. go d?y = (y dc –ady)”. zv = x2(z–p)"; .. v = (z-p)"; |