μ2 (a,x2 + bx + c) v1⁄2 + μ (ex +ƒ) pv2+gp3⁄4v2 = 0. Therefore substituting π+p for x, and developing by (13), μ2 (απ2 + bπ + c) + μ {(2αμ + e) π + (b − a) μ +ƒ} pvx μ +(μ3a+ μe+g) p3v = 0..............(b). First, let be determined so as to satisfy the equation then αμ + εμ + g = 0, μ (aπ2 + bπ + c) v„+{(2aμ+e) π+(b− a) μ+ƒ} pvz = 0. Whence, by Art. 5, v2 = Σamx (x − 1) ... (x − m + 1), the successive values of am being determined by the equation μ3 (am3 +bm + c) am+{(2aμ3+eμ) m + (b − a) μ3 + ƒμ} am2 = 0, or am (2aμ+e) m + (b − a) μ+f Represent this equation in the form and let the roots of the equation am = −ƒ(m) am-12 am2 + bm + c = 0 be a and B, then „= C {x(a) −ƒ (a+1) x(+1) +ƒ (a+1) ƒ (a + 2) x(a + 2) — &c.} +C'{x(B) -ƒ(B+1)x+1)+ƒ(B+1)ƒ(B+2)x(+2) — &c.}...(c), where generally x() = x(x-1) ... (x−p+1). One of these series will terminate whenever the value of m given by the equation (2aμ + e) m + (b − a) μ + ƒ= 0 μ exceeds by an integer either root of the equation am2 + bm + c = 0. The solution may then be completed as in the last example. Secondly, let μ be determined if possible so as to cause the second term of (b) to vanish. This gives 2aμ+e=0, (b − a) μ +ƒ= 0, μ whence, eliminating μ, we have the condition 2af + (a−b) e = 0, This being satisfied, and μ being assumed equal to e (b) becomes 2a: and is integrable in finite terms if the roots of the equation m2+m+ differ by an odd number. 11. α a Discussion of the equation (ax2 + bx + c) ▲3ux + (ex +ƒ) Aux + gux = 0. By resolution of its coefficients this equation is reducible to the form a (x − a) (x — ̈ß) ▲3u„ + e (x − y) Au2+ gu„=0.......... (a). Now let x-α=x+1 and u, v, then we have = +e(x2+a−y + 1) Av„+gv, = 0, or, dropping the accent, a (x+1)(x+a-ẞ + 1) A3v2 +e(x+a−y+1) Av ̧+gv2 = 0...(b). If from the solution of this equation v be obtained, the value of u will thence be deduced by merely changing x into x α 1. Now multiply (b) by x, and assume where Ax=1. Then, since by (20), · x (x + 1) A2v2 =π (π − 1) v2, - we have But xp, therefore substituting, and developing the coefficients we have on reduction π {α (π −a+ẞ− 1) (π − 1) + e (π-a+y− 1) + g} vx − {α (π — 1) (π — 2) + e (π − 1) + g} pv, = 0.......(c). And this is a binomial equation whose solutions in series are of the form vx = Σamx (x + 1) ... (x + m − 1), the lowest value of m being a root of the equation m {a (m − a + B − 1) (m − 1) + e (m −a+y− 1) +g}=0.....(d), corresponding to which value am is an arbitrary constant, while all succeeding values of am are determined by the law am = a (m-1) (m2) + e (m − 1) + g ́m {a (m − a + B − 1) (m − 1) + e (m − a + y − 1) +g} -Am-1° Hence the series terminates when a root of the equation a (m − 1) (m − 2) + e (m − 1) + g = 0............ (e). is equal to, or exceeds by an integer, a root of the equation (d). As a particular root of the latter equation is 0, a particular finite solution may therefore always be obtained when (e) is satisfied either by a vanishing or by a positive integral value of m. 12. The general theorem expressed by (38) admits of the following generalization, viz. F {x, ¢ (7) p′′) = II. († (7) F {w, y (7) p*} II, († (7). The ground of this extension is that the symbol π, which is here newly introduced under F, combines with the same symbol in the composition of the forms II external to F, as if π were algebraic. (11) (TT), n (11) $1 And this enables us to transform some classes of equations which are not binomial. Thus the solution of the equation ƒ。 (π) u + ƒ1 (π) $ (π) pu + ƒ2 (π) $ (π) † (π − 1) p3u= U will be made to depend upon that of the equation ƒ。(π) v+ƒ1(#) & (π) pv+ƒ1⁄2 (π) ↓ (π) † (π − 1) p3v = II ̧ 1 $ (TT). 13. While those transformations and reductions which depend upon the fundamental laws connecting π and p, and are expressed by (4), are common in their application to differential equations and to difference-equations, a marked difference exists between the two classes of equations as respects the conditions of finite solution. In differential equations where d π=0, P= ε°, there appear to be three primary integrable forms for binomial equations, viz. primary in the sense implied by the fact that every binomial equation, whatsoever its order, which admits of finite solution, is reducible to some one of the above forms by the transformations of Art. 7, founded upon the formal laws connecting π and p. In difference-equations but one primary integrable form for binomial equations is at present known, viz. and this is but a particular case of the first of the above forms for differential equations. General considerations like these may serve to indicate the path of future inquiry, 14. Many attempts have been made to accomplish the general solution of linear difference-equations with variable coefficients, but the results are in all cases so complicated as to be practically useless. It will be sufficient if we mention Spitzer (Grunert, xxxII. and xxxIII.) on the class specially considered in this chapter, viz. when the coefficients are rational integral functions of the independent variable, Libri (Crelle, XII. 234), Binet (Mémoires de l'Académie des Sciences, XIX.). There is also a brief solution by Zehfuss (Zeitschrift, III. 177), EXERCISES. 1. Of what theorem in the Differential Calculus does (20), Art. 4, constitute a generalization? 2. Solve the equation x(x+1) A3u + x▲u — n3u = 0. 3. Solve by the methods of Art. 7 the difference-equation of Ex. 1, Art. 5, supposing a to be a positive odd number. 4. Solve by the same methods the same equation, supposing a to be a negative odd number. |