and let the value thus found be substituted in d ̧u+u(U1+V10) + U2+ V ̧0=0, 1 1 3 which may be integrated by preceding methods. If U1, V1, are constant, then d0=0, and U1 — V1⁄2 0 — =0=(0—a1)(0 — a2): (c) in this case (c) becomes du + Au+B=0; by integrating which we obtain the primitive equations' 19 a1, b1, a, b, being the values of A and B when a1, a, are substituted for 0. If we have two equations each containing three variables, we may by elimination obtain three equations of the form dx+M1x + N1y+P1≈ + Q1 =0, 1 1 dry + M2x+ N2y + P¿≈ + Q2 =0, d≈ + M3x + N2y + P3≈ +Q3=0, in (a)+(b)×0, +(c)×0,=0 put u=x+01y+02≈, 2 (a) (b) (c) then by making the coefficients of y and x respectively equal to nothing we obtain d01+ (M1+ M2·01+ M ̧·02)01 − (N1 + N2·01+N3;01⁄2)=0, 3 1 2 1 d ̧02+ (M1+ M2·01+ M3.02)02 − (P1 + P2·01+P3·0 ̧)=0, whence 01 and § may be determined; substituting their values in du + (M1+ M2.01 + M3·02) u + Q1 + Q2 · O1 + Q3·02 =0, 1 2' 1 2 1 2 u, and thence the primitive equations, may be obtained. If the coefficients M1, M2, &c. are constant, the primitive equations may be determined as in the preceding instance: the same method of investigation may be applied to a greater number of equations, and to an equation of any order, in which none of the differential coefficients exceed the first degree. (L. D. C. 387-91; L. C. D. 622-4; Tr. L. 286, 7.) APPROXIMATE INTEGRATION OF DIFFERENTIAL EQUATIONS. (49.) Assume the primitive equation to be the exponents n1, n2, &c. being an ascending order, By substituting these values of y, day, &c. in the proposed differential equation, and equating corresponding indices and coefficients, we may obtain as many terms of the series as we please: the law both of the indices and coefficients is frequently evident from two or three terms. (L. D. C. 392, 3: L. C. D. 660—7.) (50.) Approximation by a continued fraction. Assume the primitive equation to be ' and substitute aœa1‚ п ̧a ̧œa‚ ̄1, &c. for y,dy, &c. respectively in the given differential equation; then by a comparison of corresponding terms, a, and n, may be determined. Then and for y, day, &c. substitute their values obtained from this equation, by which means a, and n, may be determined; the same process may be continued as far as we please. (L. D. C. 394, 5; L. C. D. 668—71.) (51.) If we have an equation of the form (52.) Lagrange's Method. Let y=4(x, C1, C2, &c. C„), in which c1, c, &c. are arbitrary constants, be the complete integral of dry+u=0: in order to extend this result to the equation day + u=av, in which u and v are functions of x, y, dy, &c. d-ly, the quantities c1, c2, &c. c,, must be considered variable; n equations may in this case. be obtained, of the form dc, y.d2c1+dc2 y.d2 c2+ &c. + dcy.dcn=0, d ̧ ̧ d ̧y.d ̧¤1 + de̱2, d2y.d2c2+ &c. + dc dxy. dxc2 = 0, From these equations the values of dc, dc, &c. dcn may be obtained dca T. v, (1) T1, T, &c. being functions of x, C1, C2, &c. which being constant when a=0, may be expanded in the form these values being substituted in the equations (1), an approximation may be obtained by neglecting all powers of a superior to the first. (L. C. D. 674, 5; Lagrange, Berlin Mem. an. 1781.) COMPARISON OF ELLIPTIC TRANSCENDENTS. (53.) Let the given equation be (a+a ̧x+a ̧à2+a ̧x3+α ̧x1) − § +(a + a1y + α2y2 + α ̧ÿ3 +α ̧y1) −13dy=0. · Suppose x and y to be functions of a new variable t, such that ·d, x = + (a + α1x + α2x2 + α ̧ Ñ3 +α ̧æ1)3, d ̧y = − (a + α1y +a‚ÿ2 + a ̧¥3 + a¡ÿ1)}; differentiating, and adding the results, we obtain d2p=a1+a2p+ 2a ̧(p2+q2) +1a ̧p (p2 + 3 q2), (1) 1 (p + q) and 1 (p − q) being substituted for x and y respectively; also d.p.dq=a ̧q+a2pq+‡a ̧q (зp2 + q2) +¦a ̧рq (p2 + q3): (2) (1) × q− (2) gives by integration a+a ̧x+a ̧æ2 +a ̧æ3 +a ̧xa]3 = (x − y) . c2 + a ̧ (x + y) +a ̧ (x+y)2]*. = in which c is an arbitrary constant. + Another integral may be obtained in a similar manner, by substituting in (2) the value of dp, whence d.q= а1+а2p+a ̧(зp2+q2)+1a ̧p(p2 + q°) If the quantity a + a1x+α2x2+a ̧æ3 +aœ1 is a perfect square, no result will be obtained; but by assuming and in the development, neglecting the powers of k superior to the first, the above integral may be reduced to that of the equation dy = { 1 — (e sin y)2} *, + {1 − (e sin x)2} {1 — Assume dx={1-(e sin x)}, then by a method analogous to the preceding we obtain or 1 — (e sin x)o1 — 1 — (e sin y)2] = c, sin (x − y), 1 — (e sin x)2 * + 1 — (e sin y)o]* = c, sin (x + y) ; either of which may be the required integral; or if ≈ represents the value of a when y=0, the integral assumes the form cosa.cosy - sina.siny. 1 — (e sin x)o |* = =COS X. (L. C. D. 690-711.) (55.) The most general form of a total differential equation containing three variables is if this equation answers the criterion of integrability (Art. 28.) its integral may be obtained under the form p(x,y,z) = c. If (1) is not immediately integrable, but may be rendered so by the introduction of a factor Q, then P(d,M — d ̧N) — N(d ̧M — d„P) + M (d ̧N — d ̧P)=0. If this equation is satisfied, the given equation may be thus integrated: let U=0 be the integral of QM + QNd ̧y=0, then U+Z=0 will be the complete integral. Z may be found by differentiating and comparing the result with the given equation. If in the proposed equation the quantities dy, d≈ exceed the first degree, these methods are applicable only when it may be resolved into factors of the form (1). (L. D. C. 4147; L. C. D. 713-21.) If the equation (1) is not integrable by the above method, a solution may be thus obtained: let Q be the factor which renders the equation M + Ndy integrable, and let the integral be U=c, ≈ being considered constant. By differentiating this last equation considering x,y,, and c, all variable, and comparing the result with (1) × Q, we obtain the equation putting c=(x), and dc=d'(x), we have which system of equations will satisfy the proposed equation (1) whatever be the form of the function p. If the equation contain four variables, a solution may be obtained by an analogous method. (L. D. C. 418-21; L. C. D. 808-13.) |