25. Examine the solution of Ex. 24, when m = 1 and when m = = 0. dy 2 27. Shew that x is an exact differential coefficient. dx2 dy 28. Shew that y® + (2xy – 1) +x 0 is an da dx2 de exact differential equation, and deduce a first integral. . 29. The equation a’y + = ( becomes integrable das (y + x*) dy by means of the factor 2.x2 dx 2xy. (Moigno, Tom. II. p. 672.) Deduce hence a first integral. 30. Deduce also the complete primitive. +1=0. X dac dic? 32. Hence deduce a singular solution of the given differential equation. 33. The complete primitive of the differential equation of the second order in Ex. 31 is required. 34. A first integral of the differential equation of the second order y - xy + 24:- (y; - ay.)' – y;' = 0 is -a) a' - (1 - 2a) ay, -a'- y'= 0, where y, stands for ) – – ) 0 dy Hence deduce the singular integral. Shew that it agrees, dx and ought to agree, with the result obtained in Art. 10. 35. Shew that the complete primitive of the above differential equation is y=x+bx + ao+bo. + y+ (-a) 36. The singular integral of the differential equation of the second order, above referred to, has been found to be 16 (1 + xo) y - 8x"y, - 16xy, + * – 16y,'=0. Ex. 2, Art. 10. Shew that this singular integral has for its complete primitive (16y + 4x* + 2*)} = x (1 + xo)! – log {(1+x?)* — «} +h, h being an arbitrary constant—and that this is a singular solution of the proposed differential equation of the second order. 37. The same singular integral has for its singular solution 16y + 4x* + ac* = 0. Prove this. Have we a right to expect that this will satisfy the differential equation of the second order? 38. By reasoning similar to that of Chap. VIII. Art. 14, shew that a singular integral of a differential equation of the form yn + f (x, y, y, ... Yn-1) = 0 will render the integrating factor of that equation infinite. •dy) 39. Differential equations of the form da2 dx/ integrated by obtaining two first integrals of the respective forms x=f(p, c), y=fi (p, c), and equating the values of p. 40. Prove the assertion in Art. 9, that a singular solution of a singular integral of a differential equation of the second order is in general no solution at all of the equation given. f( can be |