by that of the condition dp =∞, and explain this circum dy stance. 7. Shew that the singular solutions in the last two examples are of the envelope species. m 8. The differential equation y=px + (Ex. 2, Art. 5) m Ρ has y= = cx+ for its complete primitive, and y2 = 4mx for its с singular solution. Verify in this example the fundamental relation = dp log d dy dy dx dc 9. Deduce both the singular solution and the complete primitive of the differential equation y = px + √√(b2 + a2p3), and interpret each, as well as the connexion of the two, geometrically. 10. The following differential equations admit of singular solutions of the envelope species. Deduce them. 11. The equation (1 - x2) p + xy- a = 0 is satisfied by the equation y = ax. Is this a singular solution or a particular integral? ކ хр 2 Nevertheless y=0 is a particular inte is satisfied by y = 0, which also dx gral. Shew that this conclusion is in accordance with the general theorem (Art. 11). 13. The equation p(x-1)=2xy log y has a singular solution which is not of the envelope species. Determine it. L لي 14. Determine also the complete primitive in the last example, and shew how the singular solution arises. 15. The equation (p − y)2 — 2x3y (p − y) = 4x3y3 — 4x2y2 log y is satisfied by y = 0. Shew that this is a singular solution but not of the envelope species. 16. Find singular solutions of each of the following equations, and determine whether or not they are of the envelope. species. In solving the following problems, the differential equation being formed, its complete primitive as well as its singular solution is to be found and interpreted. 17. Determine a curve such that the sum of the intercepts made by the tangent on the axes of co-ordinates shall be constant and equal to a, 18. Determine a curve such that the portion of its tangent intercepted between the axes of x and y shall be constant and equal to a. 19. Find a curve always touched by the same diameter of a circle rolling along a straight line. 20. Find a curve such that the product of the perpendiculars from two fixed points upon a tangent shall be constant. (Euler. See Lagrange, Calc, des Fonctions, p. 282.) (Representing the product by k3, and the distance between the given points by 2m, making the axis of a coincide with the straight line joining them and taking for the origin of co-ordinates the middle point, the differential equation is 21. Deduce also the complete primitive of the above differential equation. 22. If the primitive of a differential equation be expressed in the form (x, y, a) = 0, the condition pressed in the form dp (x, y, a) do (x, y, a) da dy da =0 may be ex dy do (x, y, a) =∞ Hence it has sometimes been laid down that / will lead to a singular solution. Raabe, in Crelle's Journal (Veber singuläre integrale, Tom. 48), points out that this rule may fail if at the same time do (x, y, a) should become in da finite. Can it fail in any other case? 23. Exemplify Raabe's observation in the equation x+c−√/(6cy-3c2) = 0, which is the complete primitive of 3xp-6yp+x+2y= 0. At the same time shew that the singular solutions are y-x=0 and 3y+x=0. (Crelle, Ib.) 24. The complete primitive of a differential equation is (c−x + y)3 − 3 (x + y) (c − x + y)2 + 1 = 0. Representing its first member, which is rational and integral, by 6, the condition do =0 assumes the form dc 3 (c−x + y) (c− 3x − y) = 0. and Shew that ex+y=0 will not lead to a solution of the differential equation at all, while c-3x-y=0 will, explain this circumstance by a reference to Art. 4. NOTE. The reader is reminded that in all references to the general con CHAPTER IX. ON DIFFERENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST. 1. THE typical form of a differential equation of the nth order is given in Chap. I. Art. 2. We may, by solving it algebraically with respect to its highest differential coefficient, present it in the form Its genesis from a complete primitive involving n arbitrary constants has been explained, Chap. I. Art. 8. Conversely, the existence of a differential equation of the above type implies the existence of a primitive involving n arbitrary constants and no more; and a primitive possessing this character is termed complete. The converse proposition above stated, is one to which various and distinct modes of consideration point, but concerning the rigid proof of which opinion has differed. The view which appears the simplest is the following. If, as in Chap. II. Art. 2, we represent by Ap (x) the increment which the function (x) receives when receives the fixed increment Ax, and if we go on to represent by A4 (x) the increment which the function Ad (x) receives when x again receives the same fixed increment Ax, and so on, then it is evident that the values of Ap(x), A3p(x), &c., are fully determinable if the successive values of the function () in its successive states of increase are known. Thus since Ap(x) = $(x + Ax) – $(x), we have by definition A2 (x) = A {$(x + ▲x) − $ (x)} = = {Þ (x + 2Ax) – † (x + Ax)} − {† (x + Ax) − $ (x)} |