which is of Clairaut's form. The general integral is there The examples in this section are taken chiefly from Euler, Calc. Integ. Vol. 1. Sect. 111. and Vol. 11. Sect. I. Cap. 2 and 3. CHAPTER V. INTEGRATION OF DIFFERENTIAL EQUATIONS BY SERIES. THE method employed for integrating Differential Equations by series, is to assume an expression for the dependent variable in terms of the independent variable with indeterminate coefficients and indices, and then to determine them by the condition of the given equation. +2 = xa (A + A, x2+2 + A1⁄2 x2n+1 + Â ̧ Ã3n+8 + &c.) Whence we find d2 y da $3 ; = a (a− 1 ) A xa - 2 + (a + n + 2) (a + n + 1) A ̧ xa +”+&c. n and aa"y= a Axa +1 + a A1 xa+2n+2 + &c. Substituting these values in the equation, and equating to zero the coefficients of the powers of x, we have a (a1) A = 0, (a + n + 2) (a + n + 1 ) A ̧ + a A = 0, 2 (a + 2n + 4) (a + 2n + 3) A1⁄2 + a A1 The first of these is satisfied either by = 0, &c. a = 0 or a = 1. Taking a = 0 and substituting it in the other equations, we 1. 2. 3 (n + 1) (2n + 3) (3n + 5) (n + 2)3 But as this contains only one arbitrary constant A, it is not the complete solution. Let us take a = 1 and call A', A'', A2, &c. the corresponding coefficients; we then find in the same way as before A axr+3 y = ▲' { x − (n + 3) (n + 2) * 1. 2 (n + 3) (2n + 5) (n+2)2 stant. -&c.} which is another incomplete integral with one arbitrary conThe sum of these two series is the complete integral of the equation. When n = 2 both the series fail, as the denominators are then infinite: but the true integral is easily found. α. a (a− 1) + a = 0. This is a quadratic equation, which gives two values for 2r-1 The first of the preceding series will fail when n= — ber: the complete integral may however be found by the following process. Assume y = u + v log cx, where is the particular integral furnished by the series which does not fail. On substituting this value of y in the original equation we obtain the system of equations the second of which serves to determine u. Euler, Calc. Integ. Vol. 11. Chap. VII. But since we have introduced the arbitrary constant c in log car, we may assume for B, the value zero, and then we have we easily obtain a particular integral. For if we differentiate the equation r times, we have Thus any one of the coefficients in Maclaurin's Theorem is derived from the preceding one. Let the first coefficient, or the value of y when x = 0, be 4, then we find as the particular integral y = A (1 − — + + 204 12 12.22 12.22.32 12. 22. 32. 42 Let this be put equal to : then assuming and u = B+ B1 x + B2x2 + B3 No3 + &c., 1 we find by substitution in the given equation &c.) Fourier, Traité de la Chaleur, p. 372. d'y |