Thu Feb 25 19:53:44 MST 1999 aquarius% maple |\^/| Maple V Release 5.1 (WMI Campus Wide License) ._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. # ----------[ M a p l e ]---------- #interface(echo = 3); # ---------- Initialization ---------- > readlib(forget): > readlib(showtime): > _Envadditionally:= true: > _EnvAllSolutions:= true: > on; # ---------- Ordinary Difference and Differential Equations ---------- # Second order linear recurrence equationn: r(n) = (n - 1)^2 + m n O1 := r(n + 2) - 2 * r(n + 1) + r(n) = 2; r(n + 2) - 2 r(n + 1) + r(n) = 2 time = 0.02, bytes = 34098 O2 := expand(rsolve({%, r(0) = 1, r(1) = m}, r(n))); 2 -2 n + 1 + m n + n time = 0.34, bytes = 346686 # => r(n) = 3^n - 2^n [Cohen, p. 67] O3 := rsolve({r(n) = 5*r(n - 1) - 6*r(n - 2), r(0) = 0, r(1) = 1}, r(n)); n n 3 - 2 time = 0.18, bytes = 140578 # => r(n) = Fibonacci[n + 1] [Cohen, p. 83] O4 := rsolve({r(n) = r(n - 1) + r(n - 2), r(1) = 1, r(2) = 2}, r(n)); bytes used=1288008, alloc=1048384, time=1.02 1/2 / 2 \n 1/2 / 2 \n 5 |- --------| 5 |- --------| | 1/2| | 1/2| \ 1 - 5 / \ 1 + 5 / - 2/5 ------------------ + 2/5 ------------------ 1/2 1/2 1 - 5 1 + 5 time = 0.99, bytes = 1292010 # => [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] O5 := rsolve({r(n) = (1 + c - c^(n-1) - c^(n+1))/(1 - c^n)*r(n - 1) O5 := - c*(1 - c^(n-2))/(1 - c^(n-1))*r(n - 2) + 1, O5 := r(1) = 1, r(2) = (2 + 2*c + c^2)/(1 + c)}, r(n)); bytes used=2288208, alloc=1572576, time=2.16 Error, Only polynomial coefficients are handled time = 0.89, bytes = 528762 # Now, solve the ODE directly # (This has been moved ahead of the Laplace transform solution due to a bug in # forget in MapleVr5 that causes no solution to be produced.) O6 := ode:= diff(f(t), t$2) + 4*f(t) = sin(2*t); / 2 \ |d | |--- f(t)| + 4 f(t) = sin(2 t) | 2 | \dt / time = 0.04, bytes = 23974 O7 := dsolve({ode, f(0) = 0, D(f)(0) = 0}, f(t)); bytes used=3288408, alloc=2162292, time=3.45 3 f(t) = (1/8 cos(2 t) sin(2 t) - 1/4 t) cos(2 t) + 1/8 sin(2 t) time = 1.57, bytes = 1273226 O8 := simplify(%); f(t) = - 1/4 cos(2 t) t + 1/8 sin(2 t) time = 0.06, bytes = 55702 # Second order ODE with initial conditions---solve first using Laplace # transforms: f(t) = sin(2 t)/8 - t cos(2 t)/4 O9 := f(0):= 0: time = 0.02, bytes = 11718 O10 := D(f)(0):= 0: time = 0.01, bytes = 6038 O11 := inttrans[laplace](ode, t, s); 2 2 s laplace(f(t), t, s) + 4 laplace(f(t), t, s) = ------ 2 s + 4 time = 0.28, bytes = 274390 O12 := inttrans[laplace](f(t), t, s) = solve(%, inttrans[laplace](f(t), t, s)); 2 laplace(f(t), t, s) = -------------- 4 2 s + 8 s + 16 time = 0.03, bytes = 26126 O13 := inttrans[invlaplace](%, s, t); bytes used=4288600, alloc=2620960, time=5.06 f(t) = - 1/4 cos(2 t) t + 1/8 sin(2 t) time = 1.48, bytes = 764946 O14 := f(0):= 'f(0)': time = 0.01, bytes = 4094 O15 := forget(D, f); time = 0.21, bytes = 217582 O16 := ode:= 'ode': time = 0.01, bytes = 4518 # Separable equation => y(x)^2 = 2 log(x + 1) + (4 x + 3)/(x + 1)^2 + 2 A O17 := ode:= diff(y(x), x) = x^2/(y(x)*(1 + x)^3); 2 d x -- y(x) = ------------- dx 3 y(x) (1 + x) time = 0.02, bytes = 9082 O18 := dsolve(ode, y(x)); bytes used=5288820, alloc=2948580, time=6.83 2 y(x) = (2 ln(1 + x) + 4 ln(1 + x) x + 3 + 4 x + 2 _C1 x + _C1 x 2 1/2 + 2 ln(1 + x) x + _C1) /(1 + x), y(x) = - (2 ln(1 + x) + 4 ln(1 + x) x 2 2 1/2 + 3 + 4 x + 2 _C1 x + _C1 x + 2 ln(1 + x) x + _C1) /(1 + x) time = 1.66, bytes = 777862 O19 := dsolve(ode, y(x), 'implicit'); 2 1 4 y(x) + -------- - ----- - 2 ln(1 + x) - _C1 = 0 2 1 + x (1 + x) time = 0.15, bytes = 133590 O20 := y(x)^2 = solve(%, y(x)^2): time = 0.07, bytes = 43522 O21 := collect(%, {ln, _C1}): time = 0.02, bytes = 15390 O22 := lhs(%) = map(factor, rhs(%)); 2 3 + 4 x y(x) = 2 ln(1 + x) + _C1 + -------- 2 (1 + x) time = 0.02, bytes = 16470 bytes used=6289016, alloc=3210676, time=8.75 # Homogeneous equation. See Emilio O. Roxin, _Ordinary Differential # Equations_, Wadsworth Publishing Company, 1972, p. 11 # => y(x)^2 = 2 x^2 log|A x| O23 := ode:= diff(y(x), x) = y(x)/x + x/y(x); d y(x) x -- y(x) = ---- + ---- dx x y(x) time = 0.53, bytes = 13746 O24 := dsolve(ode, y(x)); 1/2 1/2 y(x) = (2 ln(x) + _C1) x, y(x) = -(2 ln(x) + _C1) x time = 0.60, bytes = 375498 O25 := dsolve(ode, y(x), 'implicit'); 2 2 2 y(x) - 2 x ln(x) - _C1 x = 0 time = 0.12, bytes = 85938 # First order linear ODE: y(x) = [A - cos(x)]/x^3 O26 := x^2*diff(y(x), x) + 3*x*y(x) = sin(x)/x; 2 /d \ sin(x) x |-- y(x)| + 3 x y(x) = ------ \dx / x time = 0.02, bytes = 7742 O27 := dsolve(%, y(x)); -cos(x) + _C1 y(x) = ------------- 3 x time = 0.29, bytes = 213326 # Exact equation => x + x^2 sin y(x) + y(x) = A [Roxin, p. 15] O28 := diff(y(x), x) = -(1 + 2*x*sin(y(x)))/(1 + x^2*cos(y(x))); d 1 + 2 x sin(y(x)) -- y(x) = - ----------------- dx 2 1 + x cos(y(x)) time = 0.02, bytes = 8538 O29 := dsolve(%, y(x)); bytes used=7289220, alloc=3210676, time=10.68 bytes used=8289400, alloc=3341724, time=12.44 2 _C1 + x + x sin(y(x)) + y(x) = 0 time = 3.66, bytes = 1983666 # Nonlinear ODE => y(x)^3/6 + A y(x) = x + B O30 := eqn:= diff(y(x), x$2) + y(x)*diff(y(x), x)^3 = 0; / 2 \ |d | /d \3 |--- y(x)| + y(x) |-- y(x)| = 0 | 2 | \dx / \dx / time = 0.01, bytes = 8542 O31 := dsolve(eqn, y(x)); bytes used=9289648, alloc=3407248, time=14.42 1/3 _C1 y(x) = %1 - 2 -----, 1/3 %1 1/3 _C1 1/2 / 1/3 _C1 \ y(x) = - 1/2 %1 + ----- + 1/2 I 3 |%1 + 2 -----|, 1/3 | 1/3| %1 \ %1 / 1/3 _C1 1/2 / 1/3 _C1 \ y(x) = - 1/2 %1 + ----- - 1/2 I 3 |%1 + 2 -----|, y(x) = _C2 1/3 | 1/3| %1 \ %1 / 3 2 2 1/2 %1 := 3 x + 3 _C2 + (8 _C1 + 9 x + 18 x _C2 + 9 _C2 ) time = 1.38, bytes = 573838 O32 := dsolve(eqn, y(x), 'implicit'); 3 1/6 y(x) + _C1 y(x) - x - _C2 = 0, y(x) = _C2 time = 0.26, bytes = 182906 # => y(x) = [3 x + sqrt(1 + 9 x^2)]^(1/3) - 1/[3 x + sqrt(1 + 9 x^2)]^(1/3) # [Pos96] O33 := dsolve({eqn, y(0) = 0, D(y)(0) = 2}, y(x)); bytes used=10289876, alloc=3472772, time=16.33 bytes used=11290088, alloc=3669344, time=18.47 bytes used=12290452, alloc=3669344, time=20.50 bytes used=13290624, alloc=3865916, time=22.45 bytes used=14290884, alloc=3931440, time=24.74 bytes used=15291172, alloc=3931440, time=26.60 bytes used=16292376, alloc=4062488, time=28.30 bytes used=17292592, alloc=4128012, time=29.92 bytes used=18292800, alloc=4128012, time=31.75 bytes used=19293008, alloc=4128012, time=33.71 bytes used=20293176, alloc=4128012, time=35.55 bytes used=21293444, alloc=4128012, time=37.80 bytes used=22295504, alloc=4193536, time=39.78 bytes used=23297096, alloc=4193536, time=42.09 bytes used=24298204, alloc=4193536, time=43.93 bytes used=25299196, alloc=4259060, time=45.72 bytes used=26299384, alloc=4259060, time=47.65 bytes used=27299836, alloc=4259060, time=49.68 2/3 2/3 2 1/2 1/2 2/3 y(x) = 1/2 (-3 %1 x - %1 (1 + 9 x ) + 3 I 3 %1 x 1/2 2/3 2 1/2 + I 3 %1 (1 + 9 x ) 1/2 2 2 1/2 1/3 1/3 2 1/2 + I 3 (18 x + 6 (1 + 9 x ) x + 1) %1 + 3 x + (1 + 9 x ) ) / 4/3 1/3 1 2/3 / %1 , y(x) = %1 - -----, y(x) = - 1/2 (3 %1 x / 1/3 %1 2/3 2 1/2 1/2 2/3 1/2 2/3 2 1/2 + %1 (1 + 9 x ) + 3 I 3 %1 x + I 3 %1 (1 + 9 x ) 1/2 2 2 1/2 1/3 1/3 2 1/2 + I 3 (18 x + 6 (1 + 9 x ) x + 1) %1 - 3 x - (1 + 9 x ) ) / 4/3 / %1 / 2 1/2 %1 := 3 x + (1 + 9 x ) time = 35.56, bytes = 18280682 # A simple parametric ODE: y(x, a) = A e^(a x) O34 := ode:= diff(y(x, a), x) = a*y(x, a); d -- y(x, a) = a y(x, a) dx time = 0.02, bytes = 9238 O35 := dsolve(ode, y(x, a)); y(x, a) = _F1(a) exp(a x) time = 0.15, bytes = 102698 O36 := ode:= 'ode': time = 0.01, bytes = 3474 # ODE with boundary conditions. This problem has nontrivial solutions # y(x) = A sin([pi/2 + n pi] x) for n an arbitrary integer O37 := assume(k <> 0): time = 0.02, bytes = 12818 O38 := dsolve({diff(y(x), x$2) + k^2*y(x) = 0, y(0) = 0, D(y)(1) = 0}, y(x)); bytes used=28300104, alloc=4259060, time=51.67 y(x) = 0 time = 0.91, bytes = 179610 O39 := k:= 'k': time = 0.00, bytes = 3334 # => y(x) = Z_v[sqrt(x)] where Z_v is an arbitrary Bessel function of order v # [Gradshteyn and Ryzhik 8.491(9)] O40 := diff(y(x), x$2) + 1/x*diff(y(x), x) + 1/(4*x)*(1 - v^2/x)*y(x) = 0; / 2 \ d | v | / 2 \ -- y(x) |1 - ----| y(x) |d | dx \ x / |--- y(x)| + ------- + 1/4 --------------- = 0 | 2 | x x \dx / time = 0.02, bytes = 10734 O41 := dsolve(%, y(x)); 1/2 1/2 y(x) = _C1 BesselY(v, x ) + _C2 BesselJ(v, x ) time = 0.35, bytes = 262782 # 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) O42 := diff(y(t), t) + a*y(t - 1) = 0; /d \ |-- y(t)| + a y(t - 1) = 0 \dt / time = 0.03, bytes = 13798 O43 := dsolve(%, y(t)); `Error, `.(`Found the indeterminate function y with different arguments`, {y(t-\ 1)}) time = 0.02, bytes = 11814 # 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) O44 := sgn:= t -> if t < 0 then -1 else 1 fi: time = 0.03, bytes = 16558 O45 := dsolve({diff(y(t), t$2) + sgn(t - TT)*y(t) = 0, y(0) = 1, D(y)(0) = 0}, y(t)); Error, cannot evaluate boolean time = 0.01, bytes = 4202 O46 := sgn:= 'sgn': time = 0.01, bytes = 3574 O47 := dsolve({diff(y(t), t$2) + sign(t - TT)*y(t) = 0, y(0) = 1, D(y)(0) = 0}, y(t)); y(t) = cos(t) time = 0.23, bytes = 153970 # 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)] O48 := eqn:= diff(i(t), t) + 2*i(t) + 5*int(i(tau), tau = 0..t) = 10*exp(-4*t); t / /d \ | |-- i(t)| + 2 i(t) + 5 | i(tau) dtau = 10 exp(-4 t) \dt / | / 0 time = 0.22, bytes = 263158 O49 := dsolve({eqn, D(i)(0) = 10}, i(t)); `Error, `.(`Found the indeterminate function i with different arguments`, {i( tau)}) time = 0.09, bytes = 47402 O50 := i(0):= 0: time = 0.01, bytes = 3946 O51 := D(i)(0):= 10: time = 0.00, bytes = 5814 O52 := inttrans[laplace](eqn, t, s); bytes used=29300300, alloc=4259060, time=53.63 laplace(i(t), t, s) 10 s laplace(i(t), t, s) + 2 laplace(i(t), t, s) + 5 ------------------- = ----- s s + 4 time = 1.09, bytes = 297918 O53 := inttrans[laplace](i(t), t, s) = solve(%, inttrans[laplace](i(t), t, s)); s laplace(i(t), t, s) = 10 --------------------- 3 2 s + 6 s + 13 s + 20 time = 0.04, bytes = 24950 O54 := inttrans[invlaplace](%, s, t); 40 40 i(t) = - -- exp(-4 t) + -- exp(-t) cos(2 t) + 5/13 exp(-t) sin(2 t) 13 13 time = 0.56, bytes = 410670 O55 := i(0):= 'i(0)': time = 0.01, bytes = 3418 O56 := forget(D, i): time = 0.21, bytes = 231198 O57 := eqn:= 'eqn': time = 0.01, bytes = 3634 # 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)] O58 := System:= [diff(x(t), t) = x(t) - y(t), diff(y(t), t) = x(t) + y(t)]; d d [-- x(t) = x(t) - y(t), -- y(t) = x(t) + y(t)] dt dt time = 0.02, bytes = 10666 O59 := sol:= dsolve({op(System)}, {x(t), y(t)}); bytes used=30300492, alloc=4324584, time=55.54 {y(t) = exp(t) (sin(t) _C1 + cos(t) _C2), x(t) = exp(t) (cos(t) _C1 - sin(t) _C2)} time = 1.32, bytes = 464786 # Check the answer O60 := map(evalb, eval(subs(sol, System))); [false, true] time = 0.03, bytes = 8262 O61 := map(evalb, simplify(eval(subs(sol, System)))); [true, true] time = 0.14, bytes = 118470 O62 := sol:= 'sol': time = 0.00, bytes = 3330 # 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. O63 := System:= [diff(x(t), t) = x(t) * (1 + cos(t)/(2 + sin(t))), O63 := diff(y(t), t) = x(t) - y(t)]; d / cos(t) \ d [-- x(t) = x(t) |1 + ----------|, -- y(t) = x(t) - y(t)] dt \ 2 + sin(t)/ dt time = 0.04, bytes = 24466 O64 := dsolve({op(System)}, {x(t), y(t)}); bytes used=31300928, alloc=4324584, time=57.59 bytes used=32311832, alloc=4390108, time=59.78 bytes used=33312016, alloc=4521156, time=61.90 bytes used=34313052, alloc=4652204, time=64.21 bytes used=35313256, alloc=4652204, time=66.16 2 exp(2 %1) _C2 (1 + tan(1/2 t) + tan(1/2 t) ) {x(t) = 5 --------------------------------------------, y(t) = (exp(-2 %1) _C1 2 1 + tan(1/2 t) 2 + exp(-2 %1) _C1 tan(1/2 t) + 2 exp(2 %1) _C2 2 / + 2 exp(2 %1) _C2 tan(1/2 t) + 3 exp(2 %1) _C2 tan(1/2 t) ) / ( / 2 1 + tan(1/2 t) )} %1 := arctan(tan(1/2 t)) time = 10.23, bytes = 4758434 O65 := simplify(convert(simplify(%, symbolic), sincos)); {y(t) = 1/2 (-_C2 cos(t) exp(2 t) + 2 _C1 + 2 _C2 sin(t) exp(2 t) + 5 _C2 exp(2 t)) exp(-t), x(t) = 5/2 (2 + sin(t)) _C2 exp(t)} time = 0.50, bytes = 449942 # Try solving this system one equation at a time O66 := factor(simplify(dsolve(System[1], x(t)), symbolic)); bytes used=36313784, alloc=4652204, time=68.10 x(t) = _C1 exp(t) (2 + sin(t)) time = 1.34, bytes = 423894 O67 := dsolve(subs(%, System[2]), y(t)); y(t) = _C1 exp(t) - 1/5 cos(t) _C1 exp(t) + 2/5 _C1 exp(t) sin(t) + exp(-t) _C2 time = 0.42, bytes = 303170 O68 := collect(collect(%, _C1), exp(t)); y(t) = (1 - 1/5 cos(t) + 2/5 sin(t)) _C1 exp(t) + exp(-t) _C2 time = 0.02, bytes = 15962 # 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) O69 := System:= {diff(x(t), t) = 2*x(t), O69 := diff(y(t), t) = -2*x(t) + y(t) - 2*z(t), O69 := diff(z(t), t) = x(t) + 3*z(t)}; d d d {-- y(t) = -2 x(t) + y(t) - 2 z(t), -- z(t) = x(t) + 3 z(t), -- x(t) = 2 x(t)} dt dt dt time = 0.05, bytes = 25394 O70 := dsolve(System, {x(t), y(t), z(t)}); bytes used=37314020, alloc=4652204, time=70.39 {x(t) = _C1 exp(2 t), z(t) = _C1 exp(3 t) - _C1 exp(2 t) + exp(3 t) _C3, y(t) = _C1 exp(t) - _C1 exp(3 t) + exp(t) _C2 + _C3 exp(t) - exp(3 t) _C3} time = 1.49, bytes = 414458 # (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)] O71 := System:= {diff(x(t), t) = y(t), diff(y(t), t) = z(t), O71 := diff(z(t), t) = -3*y(t) - 2*z(t)}; d d d {-- x(t) = y(t), -- z(t) = -3 y(t) - 2 z(t), -- y(t) = z(t)} dt dt dt time = 0.06, bytes = 27846 O72 := dsolve(System, {x(t), y(t), z(t)}); bytes used=38314244, alloc=4652204, time=72.53 1/2 1/2 1/2 {x(t) = _C1 - 2/3 _C2 exp(-t) cos(t 2 ) + 1/6 2 exp(-t) sin(t 2 ) _C2 1/2 1/2 1/2 + 2/3 _C2 - 1/6 2 exp(-t) sin(t 2 ) _C3 - 1/3 _C3 exp(-t) cos(t 2 ) + 1/3 _C3, z(t) = 1/2 exp(-t) 1/2 1/2 1/2 1/2 1/2 (-3 2 sin(t 2 ) _C2 - 2 sin(t 2 ) _C3 + 2 _C3 cos(t 2 )), y(t) = 1/2 1/2 1/2 1/2 1/2 1/2 exp(-t) (2 sin(t 2 ) _C2 + 2 _C2 cos(t 2 ) + 2 sin(t 2 ) _C3)} time = 2.62, bytes = 1474134 # (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] O73 := System:= {diff(x(t), t) = 3*x(t) - z(t), diff(y(t), t) = 2*y(t), O73 := diff(z(t), t) = x(t) + z(t)}; d d d {-- y(t) = 2 y(t), -- z(t) = x(t) + z(t), -- x(t) = 3 x(t) - z(t)} dt dt dt time = 0.02, bytes = 17158 O74 := dsolve(System, {x(t), y(t), z(t)}); {x(t) = exp(2 t) (_C1 + t _C1 - t _C3), z(t) = exp(2 t) (t _C1 + _C3 - t _C3), y(t) = _C2 exp(2 t)} time = 0.49, bytes = 350594 # 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', bytes used=39314432, alloc=4652204, time=74.42 # y(t) = -2/w [1 - cos(w t)] x0' + [4 - 3 cos(w t)] y0 + sin(w t)/w y0' O75 := System:= {diff(x(t), t$2) = 2*w*diff(y(t), t), O75 := diff(y(t), t$2) = -2*w*diff(x(t), t) + 3*w^2*y(t)}; 2 2 d /d \ d /d \ 2 {--- x(t) = 2 w |-- y(t)|, --- y(t) = -2 w |-- x(t)| + 3 w y(t)} 2 \dt / 2 \dt / dt dt time = 0.71, bytes = 19034 O76 := dsolve(System, {x(t), y(t)}); bytes used=40314612, alloc=4717728, time=76.27 {x(t) = _C1 + t _C2 + cos(w t) _C3 + sin(w t) _C4, -4 _C2 + 3 sin(w t) w _C3 - 3 cos(w t) w _C4 y(t) = - 1/6 --------------------------------------------} w time = 2.48, bytes = 1498558 O77 := System:= 'System': time = 0.01, bytes = 3330 # ---------- Quit ---------- O78 := quit bytes used=40859472, alloc=4717728, time=76.96 real 83.80 user 77.00 sys 2.40