/* =========================================== */ /* SINGLE EQUATIONS WITHOUT INITIAL CONDITIONS */ /* =========================================== */ /* FIRST ORDER EQUATIONS */ /* --------------------- */ /* Equation (1) */ eqn: (x^4-x^3)*diff(u(x),x) + 2*x^4*u(x) = x^3/3 + C; factor( ode( eqn, u(x), x ) ); /* /mathsoft/macsyma/ode/ode.so being loaded. /mathsoft/macsyma/ode/odeaux.so being loaded. Trying ode2 /mathsoft/macsyma/ode/ode2.so being loaded. /mathsoft/macsyma/library1/specfn.so being loaded. - 2 x 3 2 x 2 2 x 2 x 2 %e (2 x %e - 3 x %e + 6 c %e + 12 %c x ) (d6) u(x) = -------------------------------------------------------- 2 2 12 (x - 1) x */ /* Equation (2) */ eqn: -1/2*diff(u(x),x) + u(x) = sin(x); expand( ode( eqn, u(x), x ) ); /* Trying ode2 /mathsoft/macsyma/library1/scs.so being loaded. 4 sin(x) 2 cos(x) 2 x (d8) u(x) = -------- + -------- + %c %e 5 5 */ /* Equation (3) */ eqn: diff(y(x),x) = y(x)/(y(x)*log(y(x)) + x); ode( eqn, y(x), x ); /* Trying ode2 2 y(x) log (y(x)) - 2 x (d10) --------------------- = %c 2 y(x) */ /* Equation (4) */ eqn: 2*y(x)*diff(y(x),x)^2 - 2*x*diff(y(x),x) - y(x) = 0; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin /mathsoft/macsyma/ode/abel.so being loaded. (d12) [[], []] */ /* Equation (5) */ eqn: diff(y(x),x) + y(x) = y(x)^3*sin(x); ode( eqn, y(x), x ); /* Trying ode2 - x %e (d14) y(x) = ------------------------------------------ - 2 x 2 %e (- 2 sin(x) - cos(x)) sqrt(%c - -------------------------------) 5 */ /* Equation (6) */ eqn: diff(y(x),x) + P(x)*y(x) = Q(x)*y(x)^n; ode( eqn, y(x), x ); /* Trying ode2 / / [ [ - I p(x) dx - (n - 1) I p(x) dx 1 ] / ] ----- / [ / 1 - n (d16) y(x) = %e ((1 - n) I q(x) %e dx + %c) ] / */ ode( subst(n=1,eqn), y(x), x ); ode( subst(n=%pi,eqn), y(x), x ), n=%pi; /* ode( subs(n=%i,eqn), y(x), x ), n=%i; */ /* Equation (7) */ eqn: (x^2-1)*diff(y(x),x)^2 - 2*x*y(x)*diff(y(x),x) + y(x)^2 - 1 = 0; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin (d21) [[], []] */ /* Equation (8) */ eqn: f(x*diff(y(x),x)-y(x)) = g(diff(y(x),x)); ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin Trying int_factor /mathsoft/macsyma/ode/intfac.so being loaded. Trying odefi /mathsoft/macsyma/ode/odefi.so being loaded. Division by 0 (d23) false */ /* Equation (9) */ eqn: diff(y(x),x) = (3*x^2-y(x)^2-7)/(exp(y(x))+2*x*y(x)+1); ode( eqn, y(x), x ); /* Trying ode2 y(x) 2 3 (d25) %e + x y (x) + y(x) - x + 7 x = %c */ /* Equation (10) */ eqn: diff(y(x),x) = (2*x^3*y(x)-y(x)^4)/(x^4-2*x*y(x)^3); ode( eqn, y(x), x ); /* Trying ode2 2 x y(x) (d27) %c x = -------------------------------- 2 2 (y(x) + x) (y (x) - x y(x) + x ) */ /* Equation (11) */ eqn: diff(y(x),x)*(diff(y(x),x)+y(x)) = x*(x+y(x)); ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin 2 - x x x + 2 %c (d29) [[y(x) = - %e ((x - 1) %e - %c)], [y(x) = ---------]] 2 */ /* Equation (12) */ eqn: diff(y(x),x) = x/(x^2*y(x)^2+y(x)^5); ode( eqn, y(x), x ); /* Trying ode2 3 2 y (x) - ------- 3 2 3 (2 y (x) + 2 x + 3) %e (d31) - -------------------------------- = %c 4 */ /* Equation (13) */ eqn: y(x) = 2*x*diff(y(x),x) - a*diff(y(x),x)^3; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin (d33) [[], [], []] */ /* Equation (14) */ eqn: y(x) = 2*x*diff(y(x),x) - diff(y(x),x)^2; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin (d35) [[], []] */ /* Equation (15) */ eqn: diff(y(x),x) = exp(x)*y(x)^2 - y(x) + exp(-x); ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin Trying riccati /mathsoft/macsyma/ode/ricati.so being loaded. Trying ricsol /mathsoft/macsyma/ode/ricsol.so being loaded. /mathsoft/macsyma/share/trigsimp.so being loaded. - x - x FOUND [%i %e , - %i %e ] 2 %i x + x x %e + 2 %i %c %e (d37) y(x) = ------------------------------ 2 %i x + 2 x 2 x %i %e + 2 %c %e */ /* Equation (16) */ eqn: diff(y(x),x) = y(x)^2 - x*y(x) + 1; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin Trying riccati Trying schmidt /mathsoft/macsyma/ode/schmid.so being loaded. FOUND [x] 2 x -- %i x 2 sqrt(2) sqrt(%pi) %i x erf(-------) + 2 %e + 2 %c x sqrt(2) (d39) y(x) = ----------------------------------------------------- %i x sqrt(2) sqrt(%pi) %i erf(-------) + 2 %c sqrt(2) */ /* Equation (17) */ eqn: diff(y(x),x) = (9*x^8+1)/(y^2+1); ode( eqn, y(x), x ); /* Trying ode2 9 x + x (d41) y(x) = ------ + %c 2 y + 1 */ /* Equation (18) */ eqn: y(x) = 2*x*diff(y(x),x) + y(x)*diff(y(x),x)^2; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin (d43) [[], []] */ /* Equation (19) */ eqn: x = y(x)*diff(y(x),x)-x*diff(y(x),x)^2; ode( eqn, y(x), x ); /* Trying ode2 Trying nonlin (d45) [[], []] */ /* SECOND ORDER EQUATIONS */ /* ---------------------- */ /* Equation (20) */ eqn: diff(y(x),x,2)*(a*x+b)^2 + 4*diff(y(x),x)*(a*x+b)*a+2*y(x)*a^2 = 0; ode( eqn, y(x), x ); /* Trying ode2 %k1 x %k2 (d47) y(x) = ---------- + ---------- 2 2 (a x + b) (a x + b) */ /* Equation (21) */ /* Note: Answer yes gives a closed form of the solution */ eqn: (x^2-x)*diff(u(x),x,2) + (2*x^2+4*x-3)*diff(u(x),x)+8*x*u(x) = 1; ode( eqn, u(x), x ); /* Trying ode2 Trying desol /mathsoft/macsyma/ode/trans.so being loaded. /mathsoft/macsyma/library1/trgred.so being loaded. Trying diffsol /mathsoft/macsyma/ode/lapl.so being loaded. /mathsoft/macsyma/ode/odelinsys.so being loaded. /mathsoft/macsyma/library1/laplac.so being loaded. /mathsoft/macsyma/library1/hypgeo.so being loaded. /mathsoft/macsyma/library1/hyp.so being loaded. /mathsoft/macsyma/library1/algsys.so being loaded. /mathsoft/macsyma/library1/grobner.so being loaded. Trying odelin2 /mathsoft/macsyma/ode/odelin2.so being loaded. 2 2 2 d y d y 2 dy dy dy We solve x --- - x --- + 2 x -- + 4 x -- - 3 -- + 8 x y - 1 = 0 2 2 dx dx dx dx dx 2 dy 2 (2 x + 4 x - 3) -- d y dx 8 y First we solve --- + ------------------- + ----- = 0 2 2 x - 1 dx x - x /mathsoft/macsyma/ode/odel2pm.so being loaded. lm x Do you want to use a transformation y = u %e ? Type 1:yes or 2:no 1; - 2 x We use y = u %e du 2 -- (2 x - 8 x + 3) 2 dx u (4 x - 6) d u The result is - ------------------- - ----------- + --- = 0 2 2 2 x - x x - x dx 2 dy 2 (2 x - 8 x + 3) -- d y dx (4 x - 6) y We solve --- - ------------------- - ----------- = 0 2 2 2 dx x - x x - x u We use y = -------- 2 (x - 1) du 2 2 -- (2 x - 4 x + 3) d u dx The result is --- - ------------------- = 0 2 2 dx x - x 2 dy 2 (2 x - 4 x + 3) -- d y dx We solve --- - ------------------- = 0 2 2 dx x - x /mathsoft/macsyma/ode/odel2aux.so being loaded. which is an equation of 1st order in y' / 2 x [ (x - 1) %e y = %k1 I ------------- dx + %k2 ] 3 / x 2 x %k1 %e The solution of the last equation is --------- + %k2 2 2 x The solution of the first equation is 2 x - 2 x %k1 %e %e (--------- + %k2) 2 2 x The solution of the homogeneous equation is ------------------------- 2 (x - 1) 2 x %k1 %e %k2 (d49) u(x) = ------------------------------------ + ---------------------------- 4 2 x 3 2 x 2 2 x 2 2 x 2 x 2 x 2 x %e - 4 x %e + 2 x %e x %e - 2 x %e + %e 2 x - 3 + ----------------- 2 12 x - 24 x + 12 */ /* Equation (22) */ eqn: (x^2-x)*diff(w(x),x,2) + (1-2*x^2)*diff(w(x),x)+(4*x-2)*w(x) = 0; expand( ode( eqn, w(x), x ) ); /* Trying desol 2 x %k1 %e 2 (d51) w(x) = --------- + %k2 x 2 */ /* Equation (23) */ /* Note: Macsyma ask: Is 4 %k1 - 1 positive or negative? Both answers give two (different) implicit solutions. */ eqn: diff(y(x),x,2) - diff(y(x),x) = 2*y(x)*diff(y(x),x); ode( eqn, y(x), x ); /* Trying ode2 Is 4 %k1 - 1 positive or negative? positive; 2 y(x) + 1 2 atan(---------------) sqrt(4 %k1 - 1) (d53) ----------------------- = x + %k2 sqrt(4 %k1 - 1) Trying ode2 Is 4 %k1 - 1 positive or negative? negative; - 2 y(x) + sqrt(1 - 4 %k1) - 1 log(- ------------------------------) 2 y(x) + sqrt(1 - 4 %k1) + 1 (d55) ------------------------------------- = x + %k2 sqrt(1 - 4 %k1) */ /* Equation (24) */ eqn: diff(y(x),x,2)/y(x) - diff(y(x),x)^2/y(x)^2 - 1 + 1/y(x)^3 = 0; ode( eqn, y(x), x ); /* Trying ode2 / [ 1 sqrt(3) I ----------------------------------------- dy(x) ] 3 3 / 3 y (x) log(y(x)) + 3 %k1 y (x) + 1 sqrt(-----------------------------------) y(x) (d57) [- --------------------------------------------------------- = x + %k2, sqrt(2) / [ 1 sqrt(3) I ----------------------------------------- dy(x) ] 3 3 / 3 y (x) log(y(x)) + 3 %k1 y (x) + 1 sqrt(-----------------------------------) y(x) --------------------------------------------------------- = x + %k2] sqrt(2) */ /* Equation (25) */ eqn: diff(y(x),x,2) + 2*x*diff(y(x),x) = 2*x; ode( eqn, y(x), x ); /* Trying ode2 sqrt(%pi) %k1 erf(x) (d59) y(x) = -------------------- + x + %k2 2 */ /* Equation (26) */ eqn: 2*y(x)*diff(y(x),x,2) - diff(y(x),x)^2 = 1/3*(diff(y(x),x)-x*diff(y(x),x,2))^2; ode( eqn, y(x), x ); /* Trying ode2 Trying desol Trying diffsol Trying odelin2 2 2 2 d y 2 dy d y dy 2 We solve x (---) + (- 2 x -- - 6 y) --- + 4 (--) = 0 2 dx 2 dx dx dx /mathsoft/macsyma/ode/sings.so being loaded. Trying solfac /mathsoft/macsyma/ode/diffac.so being loaded. Trying odeseries /mathsoft/macsyma/ode/series.so being loaded. /mathsoft/macsyma/library1/combin.so being loaded. (d61) false */ /* Equation (27) */ eqn: x*diff(y(x),x,2) = 2*y(x)*diff(y(x),x); ode( eqn, y(x), x); /* Trying ode2 Trying desol Trying diffsol Trying odelin2 2 d y dy We solve x --- - 2 y -- = 0 2 dx dx Trying whittaker /mathsoft/macsyma/ode/whit.so being loaded. Trying solfac Trying odeseries (d63) false */ /* Equation (28) */ eqn: (1-x)*(y(x)*diff(y(x),x,2)-diff(y(x),x)^2)+x^2*y(x)^2 = 0; ode( eqn, y(x), x ); /* Trying ode2 Trying desol Trying diffsol Trying odelin2 2 2 d y d y dy 2 dy 2 2 2 We solve x y --- - y --- - x (--) + (--) - x y = 0 2 2 dx dx dx dx Trying solfac Trying odeseries (d65) false */ /* Equation (29) */ eqn: x*y(x)*diff(y(x),x,2) + x*diff(y(x),x)^2 + y(x)*diff(y(x),x) = 0; ode( eqn, y(x), x ); /* Trying ode2 Trying desol Trying diffsol Trying odelin2 2 d y dy 2 dy We solve x y --- + x (--) + y -- = 0 2 dx dx dx Trying whittaker Trying solfac Trying odeseries (d67) false */ /* Equation (30) */ eqn: diff(y(x),x,2)^2 - 2*diff(y(x),x)*diff(y(x),x,2) + 2*y(x)*diff(y(x),x) - y(x)^2 = 0; ode( eqn, y(x), x ); /* Trying ode2 Trying desol Trying diffsol Trying odelin2 2 2 d y 2 dy d y dy 2 We solve (---) - 2 -- --- + 2 y -- - y = 0 2 dx 2 dx dx dx Trying whittaker Trying solfac Trying odeseries (d69) false */ /* Equation (31) */ eqn: (x^3/2-x^2)*diff(y(x),x,2)+(2*x^2-3*x+1)*diff(y(x),x)+(x-1)*y(x)=0; ode( eqn, y(x), x ); /* Trying ode2 log(x) log(x - 2) log(x) log(x - 2) ------ + ---------- + 1/x - ------ - ---------- - 1/x / 2 2 2 2 [ %e (d71) y(x) = %k1 %e I --------------------------- dx ] 3 2 / x - 2 x log(x) log(x - 2) - ------ - ---------- - 1/x 2 2 + %k2 %e */ /* Equation (32) */ eqn: diff(y(x),x,2) - 2*x*diff(y(x),x) + 2*y(x) = 3; ode( eqn, y(x), x ); eqn, %, diff; expand( % ); /* Trying ode2 Trying desol 2 x %e (d77) y(x) = x (%k1 (- sqrt(%pi) %i erf(%i x) - ----) + %k2) x (c78) 2 2 x x %e %k1 %e (d78) - 2 x (%k1 (- sqrt(%pi) %i erf(%i x) - ----) + -------- + %k2) x x 2 x 2 %e x + 2 x (%k1 (- sqrt(%pi) %i erf(%i x) - ----) + %k2) + 2 %k1 %e = 3 x (c79) (d79) 0 = 3 */ /* Equation (33) */ eqn: sqrt(x)*diff(y(x),x,2)+2*x*diff(y(x),x)+3*y(x) = 0; ode( eqn, y(x), x ); /* /mathsoft/macsyma/ode/ode.so being loaded. /mathsoft/macsyma/ode/odeaux.so being loaded. /mathsoft/macsyma/ode/ode2.so being loaded. /mathsoft/macsyma/ode/trans.so being loaded. /mathsoft/macsyma/library1/trgred.so being loaded. /mathsoft/macsyma/share/trigsimp.so being loaded. /mathsoft/macsyma/ode/lapl.so being loaded. /mathsoft/macsyma/ode/odelinsys.so being loaded. /mathsoft/macsyma/library1/laplac.so being loaded. /mathsoft/macsyma/library1/hypgeo.so being loaded. /mathsoft/macsyma/library1/hyp.so being loaded. /mathsoft/macsyma/library1/algsys.so being loaded. /mathsoft/macsyma/library1/grobner.so being loaded. /mathsoft/macsyma/ode/odelin2.so being loaded. /mathsoft/macsyma/ode/odel2pm.so being loaded. the computations never stopp .... */ /* Equation (34) */ eqn: x^2*diff(y(x),x,2) + 3*x*diff(y(x),x) = 1/y(x)^3/x^4; ode( eqn, y(x), x ); /* Trying ode2 Trying desol Trying diffsol Trying odelin2 2 6 3 d y 5 3 dy We solve x y --- + 3 x y -- - 1 = 0 2 dx dx Trying whittaker Trying solfac Trying odeseries (d6) false */ /* Equation (35) */ eqn: diff(y(x),x,2) - 2/x^2*y(x) = 7*x^4 + 3*x^3; ode( eqn, y(x), x ); /* Trying ode2 6 5 3 x + 2 x 2 %k1 (d8) y(x) = ----------- + %k2 x - --- 12 3 x */ /* Equation (36) */ eqn: diff(y(x),x,2) + y(x) = csc(x); ode( eqn, y(x), x ); /* Trying ode2 /mathsoft/macsyma/library1/binoml.so being loaded. (d10) y(x) = sin(x) log(sin(x)) + %k1 sin(x) - x cos(x) + %k2 cos(x) */ /* HIGHER ORDER EQUATIONS */ /* ---------------------- */ /* Equation (37) */ eqn: diff(y(x),x,7) - 14*diff(y(x),x,6) + 80*diff(y(x),x,5) - 242*diff(y(x),x,4) + 419*diff(y(x),x,3) - 416*diff(y(x),x,2) + 220*diff(y(x),x) - 48*y(x) = 0; ode( eqn, y(x), x ); facsum( %,exp(x),exp(2*x),exp(3*x),exp(4*x) ); /* Trying ode2 /mathsoft/macsyma/library1/binoml.so being loaded. (d10) y(x) = sin(x) log(sin(x)) + %k1 sin(x) - x cos(x) + %k2 cos(x) (c11) eqn: diff(y(x),x,7) - 14*diff(y(x),x,6) + 80*diff(y(x),x,5) - 242*diff(y(x),x,4) + 419*diff(y(x),x,3) - 416*diff(y(x),x,2) + 220*diff(y(x),x) - 48*y(x) = 0; ode( eqn, y(x), x ); facsum( %,exp(x),exp(2*x),exp(3*x),exp(4 7 6 5 4 d d d d (d11) --- (y(x)) - 14 (--- (y(x))) + 80 (--- (y(x))) - 242 (--- (y(x))) 7 6 5 4 dx dx dx dx 3 2 d d d + 419 (--- (y(x))) - 416 (--- (y(x))) + 220 (-- (y(x))) - 48 y(x) = 0 3 2 dx dx dx (c12) *Trying diffsol x) ); 4 x (%k7 - 10 %k6 + 40 %k5 - 82 %k4 + 91 %k3 - 52 %k2 + 12 %k1) %e (d12) y(x) = ----------------------------------------------------------------- 108 3 x (%k7 - 11 %k6 + 47 %k5 - 101 %k4 + 116 %k3 - 68 %k2 + 16 %k1) %e - ------------------------------------------------------------------- 8 2 x %k7 x %e 2 x 2 x 2 x + ----------- - 6 %k6 x %e + 28 %k5 x %e - 65 %k4 x %e 2 2 x 159 %k3 x %e 2 x 2 x + --------------- - 49 %k2 x %e + 12 %k1 x %e 2 2 x (3 %k7 - 38 %k6 + 188 %k5 - 462 %k4 + 593 %k3 - 380 %k2 + 96 %k1) %e - ----------------------------------------------------------------------- 4 2 x 2 x 2 x 2 x 2 x %k7 x %e 13 %k6 x %e 67 %k5 x %e 175 %k4 x %e 61 %k3 x %e + ---------- - ------------- + ------------- - -------------- + ------------- 12 12 12 12 3 2 x 43 %k2 x %e 2 x - ------------- + 4 %k1 x %e + (17 %k7 - 215 %k6 + 1067 %k5 - 2645 %k4 3 x x x 17 %k7 x %e 215 %k6 x %e + 3428 %k3 - 2180 %k2 + 528 %k1) x %e /72 + ------------ - ------------- 72 72 x x x x 1067 %k5 x %e 2645 %k4 x %e 857 %k3 x %e 545 %k2 x %e + -------------- - -------------- + ------------- - ------------- 72 72 18 18 x 22 %k1 x %e + ------------ + (187 %k7 - 2329 %k6 + 11341 %k5 - 27511 %k4 + 34972 %k3 3 x - 22252 %k2 + 5808 %k1) %e /216 (c13) /mathsoft/macsyma/share/facexp.so being loaded. /mathsoft/macsyma/share/genut.so being loaded. /mathsoft/macsyma/share/index.so being loaded. (d13) y(x) = (2 (%k7 - 10 %k6 + 40 %k5 - 82 %k4 + 91 %k3 - 52 %k2 + 12 %k1) 4 x 3 x %e - 27 (%k7 - 11 %k6 + 47 %k5 - 101 %k4 + 116 %k3 - 68 %k2 + 16 %k1) %e + 54 (2 %k7 x - 24 %k6 x + 112 %k5 x - 260 %k4 x + 318 %k3 x - 196 %k2 x + 48 %k1 x - 3 %k7 + 38 %k6 - 188 %k5 + 462 %k4 - 593 %k3 + 380 %k2 - 96 %k1) 2 x 2 2 2 2 2 %e + (18 %k7 x - 234 %k6 x + 1206 %k5 x - 3150 %k4 x + 4392 %k3 x 2 2 - 3096 %k2 x + 864 %k1 x + 102 %k7 x - 1290 %k6 x + 6402 %k5 x - 15870 %k4 x + 20568 %k3 x - 13080 %k2 x + 3168 %k1 x + 187 %k7 - 2329 %k6 + 11341 %k5 x - 27511 %k4 + 34972 %k3 - 22252 %k2 + 5808 %k1) %e )/216 */ /* Equation (38) */ eqn: diff(y(x),x,4) - 4/x^2*diff(y(x),x,2) + 8/x^3*diff(y(x),x) - 8/x^4*y(x) = 0; ode( eqn, y(x), x ); /* Trying diffsol (d30) false Taking laplace transformation yields a solution laplace( eqn, x, s ); solve( %, 'laplace(y(x),x,s) ); ilt( %, s, x ); */ /* Equation (39) */ eqn: (1+x+x^2)*diff(y(x),x,3) + (3+6*x)*diff(y(x),x,2) + 6*diff(y(x),x) = 6*x; ode( eqn, y(x),x ); /* Trying diffsol (d34) false Taking laplace transf. yields a solution laplace( eqn, x, s ); solve( %,'laplace(y(x),x,s) ); solve( ilt(%,s,x ), y(x) ); */ /* Equation (40) */ eqn: (diff(y(x),x)^2+1)*diff(y(x),x,3) - 3*diff(y(x),x)*diff(y(x),x,2)^2 = 0; ode( eqn, y(x),x ); /* Trying diffsol (d36) false */ /* Equation (41) */ eqn: 3*diff(y(x),x,2) * diff(y(x),x,4) - 5*diff(y(x),x,3)^2 = 0; ode( eqn, y(x),x ); /* Trying diffsol (d38) false */ /* SPECIAL EQUATIONS */ /* ----------------- */ /* Equation (42) */ eqn: diff(y(t),t) + a*y(t-1) = 0; ode( eqn, y(t), t ); /* Trying ode2 Attempt to substitute y(t) for y in y(t - 1) Illegal substitution for operator of expression Returned to Macsyma Toplevel. */ /* Equation (43) */ eqn: diff(y(x,a),x)=a*y(x,a); ode(eqn,y(x,a),x); /* eqn: diff(y(x,a),x)=a*y(x,a); ode(eqn,y(x,a),x); d (d41) -- (y(x, a)) = a y(x, a) dx (c42) Trying ode2 a x (d42) y(x, a) = %c %e */ /* ======================================== */ /* SINGLE EQUATIONS WITH INITIAL CONDITIONS */ /* ======================================== */ /* Equation (44) */ atvalue(y(x),x=0,0)$ atvalue('diff(y(x),x),x=0,0)$ atvalue('diff(y(x),x,2),x=0,0)$ atvalue('diff(y(x),x,3),x=0,0)$ eqn: diff(y(x),x,4) = sin(x); laplace( eqn, x, s ); solve( %, 'laplace(y(x),x,s) ); ilt( %, s, x ); /* ilt( %, s, x ); 4 d (d47) --- (y(x)) = sin(x) 4 dx (c48) Proviso: Assuming s > 0. 4 1 (d48) s laplace(y(x), x, s) = ------ 2 s + 1 (c49) 1 (d49) [laplace(y(x), x, s) = -------] 6 4 s + s (c50) 3 x (d50) [y(x) = sin(x) + -- - x] 6 */ /* Equation (45) */ eqn: x*'diff(y,x,2)+'diff(y,x)+2*x*y = 0; ode( eqn,y,x ); ic2( %,x=0,y=1,'diff(y,x)=0 ); /* Trying ode2 /mathsoft/macsyma/library1/bessel.so being loaded. (d52) y = %k2 bessel_y (sqrt(2) x) + %k1 bessel_j (sqrt(2) x) 0 0 (c53) BESSEL_Y[0](0) = MINF has been generated. Returned to Macsyma Toplevel. */ /* Equation (46) */ eqn: x*'diff(y,x)^2-y^2+1=0; ode(eqn,y,x); ic1(%,x=0,y=1); /* Trying ode2 Trying nonlin /mathsoft/macsyma/ode/abel.so being loaded. 1 1 2 2 (d55) [[y = - sqrt((sqrt(-) - sqrt(-) y ) log(2 sqrt(y - 1) + 2 y) x x 1 2 1 + %c sqrt(-) y - %c sqrt(-) + 2)/sqrt(2), x x 1 1 2 2 1 2 y = sqrt((sqrt(-) - sqrt(-) y ) log(2 sqrt(y - 1) + 2 y) + %c sqrt(-) y x x x 1 - %c sqrt(-) + 2)/sqrt(2)], [y = - 1, y = 1]] x (c56) Division by 0 Returned to Macsyma Toplevel. */ /* Equation (47) */ eqn: 'diff(y,x,2) + y*'diff(y,x)^3 = 0; ode(eqn,y,x); ic2(%,x=0,y=0,'diff(y,x)=2); ratsimp(%); /* Trying ode2 3 y + 6 %k1 y (d58) ------------ = x + %k2 6 (c59) 3 2 y - 3 y (y - 1) (d59) ----------------- = x 6 (c60) 3 2 y - 3 y (d60) - ---------- = x 6 */ /* ==================== */ /* SYSTEMS OF EQUATIONS */ /* ==================== */ /* Equation (48) */ eqn1: diff(x(t),t) = -3*y(t)*z(t); eqn2: diff(y(t),t) = 3*x(t)*z(t); eqn3: diff(z(t),t) = -x(t)*y(t); odematsys( [eqn1,eqn2,eqn3], [x(t),y(t),z(t)] ); /* /mathsoft/macsyma/matrix/matsolve.so being loaded. (d64) [x(t) = x(0) - 3 ilt(laplace(3 (z(0) laplace(x(t) y(t), t, lvar) - ilt(---------------------------, lvar, t)) lvar laplace(x(t) y(t), t, lvar) laplace(x(t) (z(0) - ilt(---------------------------, lvar, t)), t, lvar) lvar ilt(-------------------------------------------------------------------------, lvar lvar, t), t, lvar)/lvar, lvar, t), y(t) = laplace(x(t) y(t), t, lvar) 3 ilt(laplace(x(t) (z(0) - ilt(---------------------------, lvar, t)), t, lvar) lvar laplace(x(t) y(t), t, lvar) /lvar, lvar, t), z(t) = z(0) - ilt(---------------------------, lvar, t)] lvar */ /* Equation (49) */ eqn1: diff(x(t),t) = a(t)*(y(t)^2-x(t)^2) + 2*b(t)*x(t)*y(t) + 2*c*x(t); eqn2: diff(y(t),t) = b(t)*(y(t)^2-x(t)^2) - 2*a(t)*x(t)*y(t) + 2*c*y(t); odematsys( [eqn1,eqn2], [x(t),y(t)] ); /* d 2 2 (d68) -- (x(t)) = a(t) (y (t) - x (t)) + 2 b(t) x(t) y(t) + 2 c x(t) dt (c69) d 2 2 (d69) -- (y(t)) = b(t) (y (t) - x (t)) - 2 a(t) x(t) y(t) + 2 c y(t) dt (c70) 2 laplace(b(t) y (t), t, lvar) (d70) [x(t) = ilt(laplace(a(t) expt(ilt(----------------------------, lvar, t) lvar - 2 c laplace(a(t) x(t) y(t), t, lvar) - 2 ilt(--------------------------------, lvar, t) lvar - 2 c 2 laplace(b(t) x (t), t, lvar) - ilt(----------------------------, lvar, t), 2), t, lvar)/(lvar - 2 c), lvar, lvar - 2 c 2 laplace(b(t) y (t), t, lvar) t) + 2 ilt(laplace(b(t) x(t) (ilt(----------------------------, lvar, t) lvar - 2 c laplace(a(t) x(t) y(t), t, lvar) - 2 ilt(--------------------------------, lvar, t) lvar - 2 c 2 laplace(b(t) x (t), t, lvar) - ilt(----------------------------, lvar, t)), t, lvar)/(lvar - 2 c), lvar, t) lvar - 2 c 2 laplace(a(t) x (t), t, lvar) 2 c t - ilt(----------------------------, lvar, t) + x(0) %e , lvar - 2 c 2 laplace(b(t) y (t), t, lvar) y(t) = ilt(----------------------------, lvar, t) lvar - 2 c laplace(a(t) x(t) y(t), t, lvar) - 2 ilt(--------------------------------, lvar, t) lvar - 2 c 2 laplace(b(t) x (t), t, lvar) - ilt(----------------------------, lvar, t)] lvar - 2 c */ /* Equation (50) */ eqn1: diff(x(t),t) = x(t)*(1+cos(t)/(2+sin(t))); eqn2: diff(y(t),t) = x(t)-y(t); odematsys( [eqn1,eqn2], [x(t),y(t)] ); /* (c73) x(t) cos(t) laplace(-----------, t, lvar) sin(t) + 2 t (d73) [x(t) = ilt(-----------------------------, lvar, t) + x(0) %e , lvar - 1 x(t) cos(t) laplace(-----------, t, lvar) t - t sin(t) + 2 x(0) %e x(0) %e y(t) = ilt(-----------------------------, lvar, t) + -------- - ----------] 2 2 2 lvar - 1 (c74) */ /* Equation (51) */ eqn1: diff(x(t),t) = 9*x(t) + 2*y(t); eqn2: diff(y(t),t) = x(t) + 8*y(t); odelinsys( [eqn1,eqn2], [x(t),y(t)] ); /* (c76) 10 t 7 t 10 t 7 t 2 x(0) %e x(0) %e x(0) %e x(0) %e (d76) [x(t) = ------------- + ----------, y(t) = ----------- - ----------] 3 3 3 3 */ /* Equation (52) */ eqn1: diff(x(t),t) - x(t) + 2*y(t) = 0; eqn2: diff(x(t),t,2) - 2*diff(y(t),t) = 2*t - cos(2*t); odelinsys( [eqn1,eqn2], [x(t),y(t)] ); /* (c79) | t/2 d | %e (34 (-- (x(t))| ) + 34 x(0) + 268) dt | | |t = 0 d | (d79) [x(t) = -------------------------------------------- - -- (x(t))| 34 dt | |t = 0 sin(2 t) 2 cos(2 t) 2 + -------- + ---------- - t - 4 t - 8, 34 17 | | t/2 d | d | %e (17 (-- (x(t))| ) + 17 x(0) + 134) -- (x(t))| + 4 dt | dt | |t = 0 |t = 0 y(t) = -------------------------------------------- - ------------------- 68 2 2 9 sin(2 t) cos(2 t) t + ---------- + -------- - -- - t] 68 34 2 */ /* Equation (53) */ eqn1: diff(y1(x),x) = -1/(x*(x^2+1))*y1(x) + 1/(x^2*(x^2+1))*y2(x) + 1/x; eqn2: diff(y2(x),x) = -x^2/(x^2+1)*y1(x) + (2*x^2+1)/(x*(x^2+1))*y2(x) + 1; odelinsys( [eqn1,eqn2], [y1(x),y2(x)] ); /* odelinsys( [eqn1,eqn2], [y1(x),y2(x)] ); 2 2 d (2 x + 1) y2(x) x y1(x) (d81) -- (y2(x)) = ---------------- - -------- + 1 dx 2 2 x (x + 1) x + 1 (c82) inf inf / / [ [ y2(x) I I laplace(------, x, lvar1) dlvar1 dlvar2 ] ] 2 / / x + 1 lvar lvar2 (d82) [y1(x) = ilt(----------------------------------------------------, lvar, lvar inf / [ y1(x) I laplace(------, x, lvar1) dlvar1 ] 2 / x + 1 lvar x) - ilt(--------------------------------------, lvar, x) lvar 1 laplace(-, x, lvar) x + ilt(-------------------, lvar, x) + y1(0), lvar inf / 2 [ d y2(x) y2(x) = ilt((I (2 (------- (laplace(------, x, lvar1))) ] 2 2 / dlvar1 x + 1 lvar y2(x) + laplace(------, x, lvar1)) dlvar1)/lvar, lvar, x) 2 x + 1 2 d y1(x) ------ (laplace(------, x, lvar)) 2 2 dlvar x + 1 - ilt(---------------------------------, lvar, x) + x + y2(0)] lvar */