-- the following was done with the help of Manuel Bronstein -- create the unknown functions y := operator y; y' := D(y x, x); y'' := D(y', x); y1 := operator y1; y2 := operator y2; y1' := D(y1 t, t); y2' := D(y2 t, t); -- firstord1 (1) eq := (x^4 - x^3) * y' + 2 * x^4 * y x = x^3/3 + C; frame0 (6) ->solve(eq,y,x) 3 2 - 2x 2x - 3x + 6C %e (6) [particular= ------------------,basis= [-----------]] 4 3 2 2 12x - 24x + 12x x - 2x + 1 -- firstord2 (2) frame0 (7) ->eq := - y' / 2 + y x = sin x; frame0 (8) ->solve(eq,y,x) 4sin(x) + 2cos(x) 2x (8) [particular= -----------------,basis= [%e ]] 5 -- firstord3 (3) frame0 (9) ->eq := y' = y x / (y x * log y x + x); frame0 (10) ->solve(eq,y,x) 2 y(x)log(y(x)) - 2x (10) ------------------- 2y(x) -- firstord4 (4) frame0 (11) ->eq := 2*(y x)*y'**2-2*x*y'-y x=0; frame0 (12) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- bernoulli (5) frame0 (12) ->eq := y' + y x = y(x)^3 * sin x; frame0 (13) ->solve(eq,y,x) 2 2 - 4y(x) sin(x) - 2y(x) cos(x) + 5 (13) --------------------------------- 2 2x 5y(x) %e -- bernoulli2 (6) frame0 (14) ->P := operator P; frame0 (15) ->Q := operator Q; frame0 (16) ->eq := y' + P x * y x = Q x * y(x)^n; frame0 (17) ->solve(eq,y,x) (17) "failed" frame0 (18) ->eq := y' + P x * y x = Q x * y(x)^2; -- the following is wrong (should contain Q) frame0 (19) ->solve(eq,y,x) 1 (19) -------------------- x ++ | P(%L)d%L ++ y(x)%e -- cannot solve for n rational frame0 (25) ->eq := y' + P x * y x = Q x * y(x)^(2/3); frame0 (26) ->solve(eq,y,x) (26) "failed" frame0 (30) ->eq := y' + P x * y x = Q x * y(x)^3; frame0 (31) ->solve(eq,y,x) 1 (31) ---------------------- x ++ | 2P(%L)d%L ++ 2 y(x) %e -- clairaut (7) frame0 (32) ->eq := (x^2-1)*y'^2-2*x*y(x)*y'+y(x)^2-1=0; frame0 (33) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- clairaut2 (8) frame0 (33) ->f := operator f; frame0 (34) ->g := operator g; frame0 (35) ->eq := f(x*y'-y(x))=g(y'); frame0 (36) ->solve(eq,y,x) >> Error detected within library code: parseODE: equation has order 0 You are being returned to the top level of the interpreter. -- exact1st (9) frame0 (36) ->eq := y' = (3 * x^2 - y(x)^2 - 7) / (exp y x + 2 * x * y x + 1); frame0 (37) ->solve(eq,y,x) y(x) 2 3 (37) %e + x y(x) + y(x) - x + 7x -- homogeneous (10) frame0 (38) ->eq := y' = (2 * x^3 * y x - y(x)^4) / (x^4 - 2 * x * y(x)^3); frame0 (39) ->solve(eq,y,x) (39) "failed" -- factor (11) frame0 (40) ->eq := y'*(y'+y(x))=x*(x+y(x)); frame0 (41) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- interchange (12) frame0 (41) ->eq := y' = x / (x^2 * y(x)^2 + y(x)^5); frame0 (42) ->solve(eq,y,x) 3 2y(x) - ------ 3 2 3 (- 2y(x) - 2x - 3)%e (42) ------------------------------ 4 -- lagrange (13) frame0 (43) ->eq := y(x)=2*x*y'-a*y'^3; frame0 (44) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- lagrange2 (14) frame0 (44) ->eq := y(x)=2*x*y'-y'^2; frame0 (45) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- riccati (15) frame0 (35) ->eq := y' = exp x * y(x)^2 - y x + exp(-x); frame0 (36) ->solve(eq,y,x) (36) "failed" -- riccati2 (16) frame0 (37) ->eq := y' = y(x)^2 - x * y x + 1; frame0 (38) ->solve(eq,y,x) 2 x - -- x 2 ++ 1 (- y(x) + x)%e | - ------- d%L + 1 ++ 2 %L - --- 2 %e (38) ----------------------------------------- 2 x - -- 2 (y(x) - x)%e -- separable (17) frame0 (39) ->eq := y' = (9 * x^8 + 1) / (y(x)^2 + 1); frame0 (40) ->solve(eq,y,x) 3 9 y(x) + 3y(x) - 3x - 3x (40) ------------------------ 3 -- solvablex (18) frame0 (41) ->eq := y(x) = 2*x*y'+y(x)*y'^2; frame0 (42) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- solvabley (19) frame0 (42) ->eq := x = y(x)*y'-x*y'^2; frame0 (43) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- SecOrderChangevar (20) frame0 (45) ->eq := y''*(a*x+b)^2+4*y'*(a*x+b)*a+2*y(x)*a^2 = 0; frame0 (48) ->solve(eq,y,x) x 2a x + b (48) [particular= 0,basis= [------------------,------------------]] 2 2 2 2 2 2 a x + 2a b x + b a x + 2a b x + b -- adjoint (21) frame0 (49) ->eq := (x^2-x)*y'' + (2*x^2+4*x-3)*y' + 8*x*y(x) = 1; frame0 (50) ->solve(eq,y,x) 3 2 - 2x 2x - 3x + 53 1 %e (50) [particular= ------------------,basis= [-------------,-----------]] 4 3 2 4 3 2 2 12x - 24x + 12x x - 2x + x x - 2x + 1 -- secondord1 (22) frame0 (51) ->eq := (x^2-x)*y'' + (1-2*x^2)*y' + (4*x-2)*y(x) = 0; frame0 (52) ->solve(eq,y,x) 2 2x (52) [particular= 0,basis= [x ,%e ]] -- autonomous (23) frame0 (53) ->eq := y''-y' = 2*y(x)*y'; frame0 (54) ->solve(eq,y,x) >> Error detected within library code: getfreelincoeff: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- autonomous2 (24) frame0 (54) ->eq := y''/y(x)-y'^2/y(x)^2-1+1/y(x)^3 = 0; frame0 (55) ->solve(eq,y,x) >> Error detected within library code: parseLODE: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- ymissing (25) frame0 (57) ->eq := y'' + 2 * x * y' = 2 * x; frame0 (58) ->solve(eq,y,x) (58) [particular= x,basis= [1,erf(x)]] -- diff (26) frame0 (59) ->eq := 2*y(x)*y''-y'^2 = (y'-x*y'')^2/3; frame0 (60) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- equidimx (27) frame0 (60) ->eq := x*y'' = 2*y(x)*y'; frame0 (61) ->solve(eq,y,x) >> Error detected within library code: getfreelincoeff: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- equidimy (28) frame0 (61) ->eq := (1-x)*(y(x)*y''-y'^2)+x^2*y(x)^2 = 0; frame0 (62) ->solve(eq,y,x) >> Error detected within library code: parseLODE: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- exact2nd (29) frame0 (62) ->eq := x*y(x)*y''+x*y'^2+y(x)*y'=0; frame0 (63) ->solve(eq,y,x) >> Error detected within library code: parseLODE: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- factoring (30) frame0 (63) ->eq := y''^2-2*y'*y''+2*y(x)*y'-y(x)^2=0; frame0 (64) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- liouvillian (31) frame0 (64) ->eq := (x^3/2-x^2)*y'' + (2*x^2-3*x+1)*y' + (x-1)*y(x) = 0; frame0 (65) ->solve(eq,y,x) 1 1 -- - - x %L x ++ %e 1 %e | ------------- d%L - - ++ +----------+ x | 4 3 %e \|%L - 2%L (65) [particular= 0,basis= [----------,----------------------------]] +-------+ +-------+ | 2 | 2 \|x - 2x \|x - 2x -- reduction (32) frame0 (68) ->eq := y'' - 2 * x * y' + 2 * y x = 3; frame0 (69) ->solve(eq,y,x) 2 x %L 3 ++ %e (69) [particular= -,basis= [x,x | ----- d%L ]] 2 ++ 2 %L -- intfactors (33) frame0 (70) ->eq := sqrt(x) * y'' + 2 * x * y' + 3 * y x = 0; frame0 (71) ->solve(eq,y,x) (71) [particular= 0,basis= []] -- scaleinv (34) frame0 (72) ->eq := x^2*y''+3*x*y'+2*y(x)=1/y(x)^3/x^4; frame0 (73) ->solve(eq,y,x) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- undet (35) frame0 (73) ->eq := y'' - 2 / x^2 * y x = 7 * x^4 + 3 * x^3; frame0 (74) ->solve(eq,y,x) 7 6 3 3 3 3x + 2x - 7x + 14 x - 1 x + 2 (74) [particular= --------------------,basis= [------,------]] 12x x x -- variation (36) frame0 (75) ->eq := y'' + y x = csc x; frame0 (76) ->solve(eq,y,x) (76) sin(x) 2 [particular= sin(x)log(----------) - sin(x)log(----------) - x cos(x), cos(x) + 1 cos(x) + 1 basis= [cos(x),sin(x)]] -- constantcoeff (37) frame0 (78) ->eq := D(y x,x,7) - 14 * D(y x,x,6) + 80 * D(y x,x,5) - 242 * D(y x,x,4) + 419 * D(y x,x,3) - 416 * y'' + 220 * y' - 48 * y x = 0; frame0 (80) ->solve(eq,y,x) 4x 3x 2x 2x x x 2 x (80) [particular= 0,basis= [%e ,%e ,%e ,x %e ,%e ,x %e ,x %e ]] -- euler (38) frame0 (81) ->eq := D(y x,x,4) - 4/x^2 * y'' + 8/x^3 * y' -8/x^4 * y x = 0; frame0 (82) ->solve(eq,y,x) [particular= 0, basis = 5 3 2 5 3 2 5 3 2 x - 5x + 5x - 1 x + 5x - 10x + 4 x - 10x + 5x + 4 [------------------, -------------------, -------------------, x x x 5 3 2 x - 10x + 20x + 4 --------------------] x ] -- exactnth (39) frame0 (83) ->eq := (1+x+x^2)*D(y x,x,3) + (3+6*x)*y'' + 6*y' = 6*x; frame0 (85) ->solve(eq,y,x) 2 x - 4 1 x + 1 (85) [particular= ------------,basis= [----------,----------,1]] 2 2 2 4x + 4x + 4 x + x + 1 x + x + 1 -- circle (40) frame0 (87) ->eq := (y'^2+1)*D(y(x),x,3)-3*y'*y''^2 = 0; frame0 (88) ->solve(eq,y,x) >> Error detected within library code: parseLODE: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- transfBernoulli (41) frame0 (88) ->eq := 3*y''*D(y(x),x,4)-5*D(y(x),x,3)^2 = 0; frame0 (89) ->solve(eq,y,x) >> Error detected within library code: parseLODE: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- delay (42) frame0 (89) ->eq := y'+a*y(x-1) = 0; frame0 (90) ->solve(eq,y,x) x ++ (90) | a y(%L - 1)d%L + y(x) ++ -- several (43) frame0 (94) ->eq := D(y(x,a),x,1) = a*y(x,a); frame0 (95) ->solve(eq,y,x) >> Error detected within library code: parseODE: equation has order 0 You are being returned to the top level of the interpreter. -- nthorder (44) frame0 (95) ->eq := D(y x,x,4) = sin x; frame0 (96) ->ini := [0, 0, 0, 0]; frame0 (97) ->solve(eq,y,x=0,ini) 3 6sin(x) + x - 6x (98) ----------------- 6 -- besselJ (45) frame0 (99) ->eq := x * y'' + y' + 2 * x * y x = 0; frame0 (100) ->solve(eq,y,x=0,[1,0]) (100) "failed" -- separ (46) frame0 (101) ->eq := x*y'^2-y(x)^2+1=0; frame0 (102) ->solve(eq,y,x=0,[1]); >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. frame0 (103) ->eq := y' = sqrt((y(x)^2-1)/x); frame0 (104) ->solve(eq,y,x=0,[1]); >> Error detected within library code: catdef: division by zero You are being returned to the top level of the interpreter. -- ic2 (47) frame0 (107) ->eq := y''+y(x)*y'^3 = 0; frame0 (108) ->solve(eq,y,x=0,[0,2]); >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- intcomb (48) -- start a new session here frame0 (1) ->x := operator x; frame0 (2) ->y := operator y; frame0 (3) ->z := operator z; frame0 (4) ->x' := D(x(t),t); frame0 (5) ->y' := D(y(t),t); frame0 (6) ->z' := D(z(t),t); frame0 (7) ->sys:=[x'=-3*y(t)*z(t),y'=3*x(t)*z(t),z'=-x(t)*y(t)]; frame0 (8) ->solve(sys,[x,y,z],t) >> Error detected within library code: getfreelincoeff: not a linear ordinary differential equation You are being returned to the top level of the interpreter. -- mriccati (49) frame0 (8) ->a := operator a; frame0 (9) ->b := operator b; frame0 (10) ->eq1 := x'=a(t)*(y(t)^2-x(t)^2)+2*b(t)*x(t)*y(t)+2*c*x(t); frame0 (11) ->eq2 := y'=b(t)*(y(t)^2-x(t)^2)-2*a(t)*x(t)*y(t)+2*c*y(t); frame0 (12) ->solve([eq1,eq2],[x,y],t) >> Error detected within library code: getlincoeff: not an appropriate ordinary differential equation You are being returned to the top level of the interpreter. -- triangular (50) frame0 (12) ->eq1 := x'=x(t)*(1+cos(t)/(2+sin(t))); frame0 (13) ->eq2 := y'=x(t)-y(t); frame0 (14) ->solve([eq1,eq2],[x,y],t) (14) "failed" -- vector (51) frame0 (15) ->eq1 := x' = 9*x(t)+2*y(t); frame0 (16) ->eq2 := y' = x(t)+8*y(t); frame0 (17) ->solve([eq1,eq2],[x,y],t) 10t 10t %e 7t 7t (17) [particular= [0,0],basis= [[%e ,-----],[%e ,- %e ]]] 2 -- highOrder (52) frame0 (18) ->eq1 := x'-x(t)+2*y(t)=0; frame0 (19) ->eq2 := D(x(t),t,2)-2*y'=2*t-cot(2*t); frame0 (20) ->solve([eq1,eq2],[x,y],t) >> Error detected within library code: solve: not a first order linear system You are being returned to the top level of the interpreter. -- inhomo (53) frame0 (20) ->eq1 := x' = -x(t)/t/(t^2+1)+y(t)/t^2/(t^2+1)+1/t; frame0 (21) ->eq2 := y' = -t^2*x(t)/(t^2+1)+(2*t^2+1)*y(t)/t/(t^2+1)+1; frame0 (22) ->solve([eq1,eq2],[x,y],t) 1 2 (22) [particular= [log(t) - 1,t log(t) - t],basis= [[-,- t ],[1,t]]] t -- bronstein s := sqrt(-222); a0 := 104/25*x^10+(274/25-22/15*s)*x^8+(7754/75-68/15*s)*x^6; a0 := a0+(11248/75-194/15*s)*x^4+(29452/75-296/5*s)*x^2-10952/5-148/3*s; a2 := x^12+2*x^10+151/3*x^8+296/3*x^6+5920/9*x^4+10952/9*x^2+5476/9; eq := a2*y'' - a0*y(x); frame0 (23) ->solve(eq,y,x) -- moussiaux frame0 (2) ->solve(15*D(y(x),x)+24*y(x)^2=7*x^(-8/3),y,x) (3) "failed"