Tue Feb 23 17:18:04 MST 1999 aquarius% macsyma Starting Macsyma math engine with no window system... /opt/local/macsyma_422/macsyma.422 local This is Macsyma 422.0 for Sparc (Solaris 2.x) computers. Copyright (c) 1982 - 1998 Macsyma Inc. All rights reserved. Portions copyright (c) 1982 Massachusetts Institute of Technology. All rights reserved. Type "DESCRIBE(TRADE_SECRET);" to see important legal notices. Type "HELP();" for more information. /aquarius/data2/opt/local/macsyma_422/system/init.lsp being loaded. /aquarius/home/wester/macsyma-init.lsp being loaded. (c1) (c2) /* ----------[ M a c s y m a ]---------- */ /* ---------- Initialization ---------- */ symbol_display_case: lower_case$ Time= 0 msecs (c3) showtime: all$ Time= 0 msecs (c4) prederror: false$ Time= 0 msecs (c5) /* ---------- Ordinary Difference and Differential Equations ---------- */ /* Second order linear recurrence equation: r(n) = (n - 1)^2 + m n */ differenceq(r[n + 2] - 2 * r[n + 1] + r[n] = 2, r[n], [r[0] = 1, r[1] = m]); /aquarius/data2/opt/local/macsyma_422/share/differ.so being loaded. /aquarius/data2/opt/local/macsyma_422/share/recur.so being loaded. Time= 430 msecs 2 (d5) r = n + (m - 2) n + 1 n (c6) /* => r(n) = 3^n - 2^n [Cohen, p. 67] */ errcatch(differenceq(r[n] = 5*r[n - 1] - 6*r[n - 2], r[n], [r[0] = 0, r[1] = 1]))$ Improper equation for DIFFERENCEQ: errexp1 Time= 20 msecs (c7) differenceq(r[n + 2] = 5*r[n + 1] - 6*r[n], r[n], [r[0] = 0, r[1] = 1]); Time= 60 msecs n n (d7) r = 3 - 2 n (c8) /* => r(n) = Fibonacci[n + 1] [Cohen, p. 83] */ errcatch(differenceq(r[n] = r[n - 1] + r[n - 2], r[n], [r[1] = 1, r[2] = 2]))$ Improper equation for DIFFERENCEQ: errexp1 Time= 10 msecs (c9) expr: differenceq(r[n + 2] = r[n + 1] + r[n], r[n], [r[1] = 1, r[2] = 2]); Time= 290 msecs n (sqrt(5) + 1) (- sqrt(5) r + 5 r + 2 sqrt(5)) 0 0 (d9) r = ------------------------------------------------ n n 10 2 n n (sqrt(5) - 1) (- sqrt(5) r - 5 r + 2 sqrt(5)) (- 1) 0 0 - ------------------------------------------------------- n 10 2 (c10) solve(2 = subst(n = 2, rhs(%))); Time= 30 msecs (d10) [r = 1] 0 (c11) expr: factorsum(subst(%, expr)); Time= 60 msecs n n n (sqrt(5) - 5) (sqrt(5) - 1) (- 1) - (sqrt(5) + 1) (sqrt(5) + 5) (d11) r = - ------------------------------------------------------------------ n n 10 2 (c12) subst(%phi = (1 + sqrt(5))/2, fibtophi(fib(n+1))); /aquarius/data2/opt/local/macsyma_422/library1/combin.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/combin.so being loaded. Time= 310 msecs n + 1 (sqrt(5) + 1) sqrt(5) + 1 n + 1 ------------------ - (1 - -----------) n + 1 2 2 (d12) ------------------------------------------- sqrt(5) (c13) ratsimp(rhs(expr) - %); Time= 10 msecs (d13) 0 (c14) remvalue(expr)$ Time= 0 msecs (c15) /* => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)] [Joan Z. Yu and Robert Israel in sci.math.symbolic] */ eqn: r[n] = (1 + c - c^(n-1) - c^(n+1))/(1 - c^n)*r[n - 1] - c*(1 - c^(n-2))/(1 - c^(n-1))*r[n - 2] + 1; Time= 10 msecs n + 1 n - 1 n - 2 (- c - c + c + 1) r c (1 - c ) r n - 1 n - 2 (d15) r = ---------------------------------- - --------------------- + 1 n n n - 1 1 - c 1 - c (c16) errcatch(differenceq(eqn, r[n], [r[1] = 1, r[2] = (2 + 2*c + c^2)/(1 + c)]))$ Improper equation for DIFFERENCEQ: errexp1 Time= 30 msecs (c17) remvalue(eqn)$ Time= 0 msecs (c18) /* Second order ODE with initial conditions---solve first using Laplace transforms: f(t) = sin(2 t)/8 - t cos(2 t)/4 */ atvalue(f(t), t = 0, 0)$ Time= 0 msecs (c19) atvalue(diff(f(t), t), t = 0, 0)$ Time= 0 msecs (c20) printprops(all, atvalue)$ | d | --- (f(@1))| = 0 d@1 | |@1 = 0 f(0) = 0 Time= 10 msecs (c21) ode: diff(f(t), t, 2) + 4*f(t) = sin(2*t); Time= 10 msecs 2 d (d21) --- (f(t)) + 4 f(t) = sin(2 t) 2 dt (c22) laplace(ode, t, s); /aquarius/data2/opt/local/macsyma_422/library1/laplac.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/hypgeo.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/hyp.so being loaded. Proviso: Assuming s > 0. Time= 620 msecs 2 2 (d22) s laplace(f(t), t, s) + 4 laplace(f(t), t, s) = ------ 2 s + 4 (c23) solve(%, 'laplace(f(t), t, s)); Time= 20 msecs 2 (d23) [laplace(f(t), t, s) = --------------] 4 2 s + 8 s + 16 (c24) ilt(%[1], s, t); Time= 260 msecs sin(2 t) t cos(2 t) (d24) f(t) = -------- - ---------- 8 4 (c25) /* Now, solve the ODE directly */ subst(f(t) = f, ode); Time= 0 msecs 2 d f (d25) --- + 4 f = sin(2 t) 2 dt (c26) ode(%, f, t); /aquarius/data2/opt/local/macsyma_422/ode/ode.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/odeaux.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/ode2.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/binoml.so being loaded. Time= 870 msecs t cos(2 t) (d26) f = %k1 sin(2 t) - ---------- + %k2 cos(2 t) 4 (c27) ode_ibc(%, t = 0, f = 0, t = 0, 'diff(f, t) = 0); Time= 30 msecs sin(2 t) t cos(2 t) (d27) f = -------- - ---------- 8 4 (c28) remvalue(ode)$ Time= 0 msecs (c29) /* Separable equation => y(x)^2 = 2 log(x + 1) + (4 x + 3)/(x + 1)^2 + 2 A */ 'diff(y, x) = x^2/(y*(1 + x)^3); Time= 10 msecs 2 dy x (d29) -- = ---------- dx 3 (x + 1) y (c30) ode(%, y, x); Time= 120 msecs 2 2 y (2 x + 4 x + 2) log(x + 1) + 4 x + 3 (d30) -- = ------------------------------------- + %c 2 2 2 x + 4 x + 2 (c31) lhs(%) = map('factorsum, rhs(%)); Time= 20 msecs 2 2 y 2 (x + 1) log(x + 1) + 4 x + 3 (d31) -- = ------------------------------- + %c 2 2 2 (x + 1) (c32) /* Homogeneous equation. See Emilio O. Roxin, _Ordinary Differential Equations_, Wadsworth Publishing Company, 1972, p. 11 => y(x)^2 = 2 x^2 log|A x| */ 'diff(y, x) = y/x + x/y; Time= 0 msecs dy y x (d32) -- = - + - dx x y (c33) ode(%, y, x); Time= 350 msecs 2 2 2 y - x log(x ) (d33) --------------- = %c 2 2 x (c34) /* First order linear ODE: y(x) = [A - cos(x)]/x^3 */ x^2*'diff(y, x) + 3*x*y = sin(x)/x; Time= 0 msecs 2 dy sin(x) (d34) x -- + 3 x y = ------ dx x (c35) ode(%, y, x); Time= 70 msecs %c - cos(x) (d35) y = ----------- 3 x (c36) /* Exact equation => x + x^2 sin y(x) + y(x) = A [Roxin, p. 15] */ 'diff(y, x) = -(1 + 2*x*sin(y))/(1 + x^2*cos(y)); Time= 0 msecs dy 2 x sin(y) + 1 (d36) -- = - -------------- dx 2 x cos(y) + 1 (c37) ode(%, y, x); Time= 100 msecs 2 (d37) x sin(y) + y + x = %c (c38) /* Nonlinear ODE => y(x)^3/6 + A y(x) = x + B */ 'diff(y, x, 2) + y*'diff(y, x)^3 = 0; Time= 10 msecs 2 d y dy 3 (d38) --- + y (--) = 0 2 dx dx (c39) ode(%, y, x); Time= 140 msecs 3 y + 6 %k1 y (d39) ------------ = x + %k2 6 (c40) /* => y(x) = [3 x + sqrt(1 + 9 x^2)]^(1/3) - 1/[3 x + sqrt(1 + 9 x^2)]^(1/3) [Pos96] */ errcatch(ode_ibc(%, x = 0, y = 0, x = 0, 'diff(y, x) = 2)); dy Cannot impose -- = 2 on errexp1 dx Time= 10 msecs (d40) [] (c41) /* A simple parametric ODE: y(x, a) = A e^(a x) */ diff(y(x, a), x) = a*y(x, a); Time= 0 msecs d (d41) -- (y(x, a)) = a y(x, a) dx (c42) ode(%, y(x, a), x); Time= 30 msecs a x (d42) y(x, a) = %c %e (c43) /* ODE with boundary conditions. This problem has nontrivial solutions y(x) = A sin([pi/2 + n pi] x) for n an arbitrary integer */ assume(not(equal(k, 0)))$ Time= 0 msecs (c44) ode('diff(y, x, 2) + k^2*y = 0, y, x); Time= 30 msecs (d44) y = %k1 sin(k x) + %k2 cos(k x) (c45) ode_ibc(%, x = 0, y = 0, x = 1, 'diff(y, x) = 0); Time= 30 msecs (d45) y = 0 (c46) forget(not(equal(k, 0)))$ Time= 0 msecs (c47) /* => y(x) = Z_v[sqrt(x)] where Z_v is an arbitrary Bessel function of order v [Gradshteyn and Ryzhik 8.491(9)] */ eqn: 'diff(y, x, 2) + 1/x*'diff(y, x) + 1/(4*x)*(1 - v^2/x)*y = 0; Time= 10 msecs 2 dy v 2 -- (1 - --) y d y dx x (d47) --- + -- + ---------- = 0 2 x 4 x dx (c48) declare(v, noninteger)$ Time= 0 msecs (c49) ode(eqn, y, x); /aquarius/data2/opt/local/macsyma_422/ode/lapl.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/odelins.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/odelin2.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/odel2pm.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/odel2aux.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/bessel.so being loaded. Time= 2500 msecs (d49) y = %k1 bessel_j (sqrt(x)) + %k2 bessel_j (sqrt(x)) v - v (c50) remove(v, noninteger)$ Time= 0 msecs (c51) declare(v, integer)$ Time= 0 msecs (c52) ode(eqn, y, x); Time= 1100 msecs (d52) y = %k2 bessel_y (sqrt(x)) + %k1 bessel_j (sqrt(x)) v v (c53) remove(v, integer)$ Time= 0 msecs (c54) /* Delay (or mixed differential-difference) equation. See Daniel Zwillinger, _Handbook of Differential Equations_, Second Edition, Academic Press, Inc., 1992, p. 210 => y(t) = y0 sum((-a)^n (t - n + 1)^n/n!, n = 0..floor(t) + 1) */ 'diff(y(t), t) + a*y(t - 1) = 0; Time= 0 msecs d (d54) -- (y(t)) + a y(t - 1) = 0 dt (c55) ode(%, y(t), t); Time= 50 msecs / [ (d55) a I y(t - 1) dt + y(t) = %c ] / (c56) /* Discontinuous ODE [Zwillinger, p. 221] => y(t) = cosh t (0 <= t < T) (sin T cosh T + cos T sinh T) sin t + (cos T cosh T - sin T sinh T) cos t (T <= t) */ sgn(t):= if t < 0 then -1 else 1$ Time= 0 msecs (c57) ode(diff(y(t), t, 2) + sgn(t - TT)*y(t) = 0, y(t), t); The argument to FIRST is empty: [] /aquarius/data2/opt/local/macsyma_422/ode/sings.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/diffac.so being loaded. /aquarius/data2/opt/local/macsyma_422/ode/series.so being loaded. Time= 1130 msecs (d57) false (c58) /*ode_ibc(%, t = 0, y(t) = 1, t = 0, 'diff(y(t), t) = 0);*/ remfunction(sgn)$ Time= 0 msecs (c59) assume(pnz > 0)$ Time= 10 msecs (c60) ode(diff(y(t), t, 2) + sign(t - TT)*y(t) = 0, y(t), t); Time= 80 msecs (d60) y(t) = %k1 sin(sqrt(pnz) t) + %k2 cos(sqrt(pnz) t) (c61) errcatch(ode_ibc(%, t = 0, y(t) = 1, t = 0, 'diff(y(t), t) = 0)); Cannot impose y(t) = 1 on errexp1 Time= 10 msecs (d61) [] (c62) forget(pnz > 0)$ Time= 0 msecs (c63) /* Integro-differential equation. See A. E. Fitzgerald, David E. Higginbotham and Arvin Grabel, _Basic Electrical Engineering_, Fourth Edition, McGraw-Hill Book Company, 1975, p. 117. => i(t) = 5/13 [-8 e^(-4 t) + e^(-t) (8 cos 2 t + sin 2 t)] */ eqn: diff(i(t), t) + 2*i(t) + 5*integrate(i(tau), tau, 0, t) = 10*%e^(-4*t); Time= 10 msecs t / [ d - 4 t (d63) 5 I i(tau) dtau + -- (i(t)) + 2 i(t) = 10 %e ] dt / 0 (c64) ode(eqn, i(t), t); /aquarius/data2/opt/local/macsyma_422/library1/scs.so being loaded. Time= 220 msecs t / / - 2 t [ 2 t [ - 2 t (d64) i(t) = %e (- 5 I %e I i(tau) dtau dt - 5 %e + %c) ] ] / / 0 (c65) atvalue(i(t), t = 0, 0)$ Time= 10 msecs (c66) atvalue(diff(i(t), t), t = 0, 10)$ Time= 0 msecs (c67) laplace(eqn, t, s); Proviso: Assuming s > 0. Time= 50 msecs 5 laplace(i(t), t, s) (d67) s laplace(i(t), t, s) + --------------------- + 2 laplace(i(t), t, s) = s 10 ----- s + 4 (c68) solve(%, 'laplace(i(t), t, s)); Time= 30 msecs 10 s (d68) [laplace(i(t), t, s) = ---------------------] 3 2 s + 6 s + 13 s + 20 (c69) ilt(%[1], s, t); Time= 30 msecs - 4 t - t 5 sin(2 t) 40 cos(2 t) 40 %e (d69) i(t) = %e (---------- + -----------) - ---------- 13 13 13 (c70) remvalue(eqn)$ Time= 0 msecs (c71) /* System of two linear, constant coefficient ODEs: x(t) = e^t [A cos(t) - B sin(t)], y(t) = e^t [A sin(t) + B cos(t)] */ system: [diff(x(t), t) = x(t) - y(t), diff(y(t), t) = x(t) + y(t)]; Time= 0 msecs d d (d71) [-- (x(t)) = x(t) - y(t), -- (y(t)) = y(t) + x(t)] dt dt (c72) expand(odelinsys(system, [x(t), y(t)])); Time= 160 msecs t t (d72) [x(t) = x(0) %e cos(t) - y(0) %e sin(t), t t y(t) = x(0) %e sin(t) + y(0) %e cos(t)] (c73) /* Check the answer */ ratsimp(ev(system, %, diff)); Time= 60 msecs t t (d73) [(- y(0) - x(0)) %e sin(t) + (x(0) - y(0)) %e cos(t) = t t (- y(0) - x(0)) %e sin(t) + (x(0) - y(0)) %e cos(t), t t (x(0) - y(0)) %e sin(t) + (y(0) + x(0)) %e cos(t) = t t (x(0) - y(0)) %e sin(t) + (y(0) + x(0)) %e cos(t)] (c74) map(lambda([q], is(equal(lhs(q), rhs(q)))), %); Time= 0 msecs (d74) [true, true] (c75) /* Triangular system of two ODEs: x(t) = A e^t [sin(t) + 2], y(t) = A e^t [5 - cos(t) + 2 sin(t)]/5 + B e^(-t). See Nicolas Robidoux, ``Does Axiom Solve Systems of O.D.E.'s Like Mathematica?'', LA-UR-93-2235, Los Alamos National Laboratory, Los Alamos, New Mexico. */ system: [diff(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), diff(y(t), t) = x(t) - y(t)]; Time= 10 msecs d cos(t) d (d75) [-- (x(t)) = x(t) (---------- + 1), -- (y(t)) = x(t) - y(t)] dt sin(t) + 2 dt (c76) odelinsys(system, [x(t), y(t)]); Time= 1410 msecs t (d76) [x(t) = %k1 %e (sin(t) + 2), y(t) = 2 t - t %k1 %e (2 sin(t) - cos(t)) 2 t %e (----------------------------- + %k1 %e + %k2)] 5 (c77) /* Try solving this system one equation at a time */ subst(%c = %c1, factor(ode(system[1], x(t), t))); Time= 830 msecs t (d77) x(t) = %c1 %e (sin(t) + 2) (c78) map('factorsum, ode(subst(%, system[2]), y(t), t)); Time= 180 msecs - t 2 t %e (%c1 %e (2 sin(t) - cos(t) + 5) + 5 %c) (d78) y(t) = ------------------------------------------------ 5 (c79) /* 3 x 3 linear system with constant coefficients: (1) real distinct characteristic roots (= 2, 1, 3) [Roxin, p. 109] => x(t) = A e^(2 t), y(t) = B e^t + C e^(3 t), z(t) = -A e^(2 t) - C e^(3 t) */ system: [diff(x(t), t) = 2*x(t), diff(y(t), t) = -2*x(t) + y(t) - 2*z(t), diff(z(t), t) = x(t) + 3*z(t)]; Time= 0 msecs d d (d79) [-- (x(t)) = 2 x(t), -- (y(t)) = - 2 z(t) + y(t) - 2 x(t), dt dt d -- (z(t)) = 3 z(t) + x(t)] dt (c80) odelinsys(system, [x(t), y(t), z(t)]); Time= 510 msecs 2 t 3 t (d80) [x(t) = x(0) %e , y(t) = (- z(0) - x(0)) %e t 3 t 2 t + (z(0) + y(0) + x(0)) %e , z(t) = (z(0) + x(0)) %e - x(0) %e ] (c81) /* (2) complex characteristic roots (= 0, -1 +- sqrt(2) i) [Roxin, p. 111] => x(t) = A + e^(-t)/3 [-(B + sqrt(2) C) cos(sqrt(2) t) + (sqrt(2) B - C) sin(sqrt(2) t)], y(t) = e^(-t) [B cos(sqrt(2) t) + C sin(sqrt(2) t)], z(t) = e^(-t) [(-B + sqrt(2) C) cos(sqrt(2) t) -(sqrt(2) B + C) sin(sqrt(2) t)] */ system: [diff(x(t), t) = y(t), diff(y(t), t) = z(t), diff(z(t), t) = -3*y(t) - 2*z(t)]; Time= 0 msecs d d d (d81) [-- (x(t)) = y(t), -- (y(t)) = z(t), -- (z(t)) = - 2 z(t) - 3 y(t)] dt dt dt (c82) odelinsys(system, [x(t), y(t), z(t)]); Time= 270 msecs 2 (z(0) + 2 y(0)) 2 (2 z(0) + y(0)) (----------------- - -----------------) sin(sqrt(2) t) - t 3 3 (d82) [x(t) = %e (------------------------------------------------------ 2 sqrt(2) (z(0) + 2 y(0)) cos(sqrt(2) t) z(0) + 2 y(0) + 3 x(0) - ------------------------------) + ----------------------, 3 3 - t (2 (z(0) + 2 y(0)) - 2 y(0)) sin(sqrt(2) t) y(t) = %e (------------------------------------------- 2 sqrt(2) - t (- 2 z(0) - 6 y(0)) sin(sqrt(2) t) + y(0) cos(sqrt(2) t)), z(t) = %e (---------------------------------- 2 sqrt(2) + z(0) cos(sqrt(2) t))] (c83) ratsimp(%); Time= 70 msecs - t (d83) [x(t) = - %e ((z(0) - y(0)) sin(sqrt(2) t) + (sqrt(2) z(0) + 2 sqrt(2) y(0)) cos(sqrt(2) t) t + (- sqrt(2) z(0) - 2 sqrt(2) y(0) - 3 sqrt(2) x(0)) %e )/(3 sqrt(2)), - t %e ((z(0) + y(0)) sin(sqrt(2) t) + sqrt(2) y(0) cos(sqrt(2) t)) y(t) = ------------------------------------------------------------------, sqrt(2) - t %e ((z(0) + 3 y(0)) sin(sqrt(2) t) - sqrt(2) z(0) cos(sqrt(2) t)) z(t) = - --------------------------------------------------------------------] sqrt(2) (c84) /* (3) multiple characteristic roots (= 2, 2, 2) [Roxin, p. 113] => x(t) = e^(2 t) [A + C (1 + t)], y(t) = B e^(2 t), z(t) = e^(2 t) [A + C t] */ system: [diff(x(t), t) = 3*x(t) - z(t), diff(y(t), t) = 2*y(t), diff(z(t), t) = x(t) + z(t)]; Time= 260 msecs d d d (d84) [-- (x(t)) = 3 x(t) - z(t), -- (y(t)) = 2 y(t), -- (z(t)) = z(t) + x(t)] dt dt dt (c85) odelinsys(system, [x(t), y(t), z(t)]); Time= 230 msecs 2 t 2 t 2 t 2 t (d85) [x(t) = - z(0) t %e + x(0) t %e + x(0) %e , y(t) = y(0) %e , 2 t 2 t 2 t z(t) = - z(0) t %e + x(0) t %e + z(0) %e ] (c86) /* x(t) = x0 + [4 sin(w t)/w - 3 t] x0' [Rick Niles] + 6 [w t - sin(w t)] y0 + 2/w [1 - cos(w t)] y0', y(t) = -2/w [1 - cos(w t)] x0' + [4 - 3 cos(w t)] y0 + sin(w t)/w y0' */ system: [diff(x(t), t, 2) = 2*w*diff(y(t), t), diff(y(t), t, 2) = -2*w*diff(x(t), t) + 3*w^2*y(t)]; Time= 10 msecs 2 2 d d d 2 d (d86) [--- (x(t)) = 2 (-- (y(t))) w, --- (y(t)) = 3 y(t) w - 2 (-- (x(t))) w] 2 dt 2 dt dt dt (c87) odelinsys(system, [x(t), y(t)]); Time= 570 msecs | d | (4 (-- (x(t))| ) - 6 y(0) w) sin(t w) dt | |t = 0 (d87) [x(t) = ----------------------------------------- w | | d | d | 2 (-- (y(t))| ) cos(t w) x(0) w + 2 (-- (y(t))| ) dt | dt | |t = 0 |t = 0 - ---------------------------- + ---------------------------- + 6 y(0) t w w w | d | (-- (y(t))| ) sin(t w) | dt | d | |t = 0 - 3 t (-- (x(t))| ), y(t) = -------------------------- dt | w |t = 0 | | d | d | (3 y(0) w - 2 (-- (x(t))| )) cos(t w) 4 y(0) w - 2 (-- (x(t))| ) dt | dt | |t = 0 |t = 0 - ----------------------------------------- + ------------------------------] w w (c88) remvalue(system)$ Time= 0 msecs (c89) /* ---------- Quit ---------- */ quit(); Bye. real 22.90 user 15.63 sys 2.68