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Online Algebra Calculator
Many of users do not use powerful Maxima computer algebra system for systematic work, but for ad hoc algebraic calculations – equations, functions, matrixes, etc. only. To do this job, it is not useful to download the whole application from a web site.
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(%i1)eqn:(Dy)^2+y^2+4*y*exp(x)-x*sin(x); 2 x 2 (%o1) y + 4 %e y - x sin(x) + Dy (%i2) h:diff(eqn,y); x (%o2) 2 y + 4 %e (%i3) h1:diff(eqn,Dy); (%o3) 2 Dy (%i4) euler:h-(diff(h1, x) + diff(h1, y)*Dy+diff(h1, Dy)*'diff(Dy, x))=0; x dDy (%o4) 2 y + 4 %e - 2 --- = 0 dx (%i5) eqn1:subst(Dy='diff(y, x), euler); 2 d y x (%o5) - 2 --- + 2 y + 4 %e = 0 2 dx (%i6) sol:ode2(eqn1, y, x); x (2 x - 1) %e x - x (%o6) y = ------------- + %k1 %e + %k2 %e 2 (%i7) extr:bc2(sol, x=-1, y=1, x=1, y=3); x 3 2 x 2 - x (2 x - 1) %e (%e - 5 %e - 5 %e - 3) %e (%e + 3 %e) %e (%o7) y = ------------- - ---------------------------- + ------------------ 2 3 2 3 2 2 %e + 2 %e + 2 %e + 2 %e + %e + %e + 1 (%i8) x1:-1; (%o8) - 1 (%i9) y1:1; (%o9) 1 (%i10) x2:1; (%o10) 1 (%i11) y2:3; (%o11) 3 (%i12) b0:y1+(y2-y1)*(x-x1)/(x2-x1); (%o12) x + 2 (%i13) b(k):=sin(k*%pi*(x-x1)/(x2-x1)); k %pi (x - x1) (%o13) b(k) := sin(--------------) x2 - x1 (%i14) yr:b0+a1*b(1)+a2*b(2)+a3*b(3)+a4*b(4)+a5*b(5)+a6*b(6); 5 %pi (x + 1) (%o14) a6 sin(3 %pi (x + 1)) + a5 sin(-------------) + a4 sin(2 %pi (x + 1)) 2 3 %pi (x + 1) %pi (x + 1) + a3 sin(-------------) + a2 sin(%pi (x + 1)) + a1 sin(-----------) + x + 2 2 2 (%i15) Dyr:diff(yr,x); 5 %pi (x + 1) 5 %pi a5 cos(-------------) 2 (%o15) 3 %pi a6 cos(3 %pi (x + 1)) + --------------------------- 2 3 %pi (x + 1) 3 %pi a3 cos(-------------) 2 + 2 %pi a4 cos(2 %pi (x + 1)) + --------------------------- 2 %pi (x + 1) %pi a1 cos(-----------) 2 + %pi a2 cos(%pi (x + 1)) + ----------------------- + 1 2 (%i16) Fyr:subst([y=yr, Dy=Dyr],eqn); 5 %pi (x + 1) (%o16) expt(a6 sin(3 %pi (x + 1)) + a5 sin(-------------) 2 3 %pi (x + 1) + a4 sin(2 %pi (x + 1)) + a3 sin(-------------) + a2 sin(%pi (x + 1)) 2 %pi (x + 1) x + a1 sin(-----------) + x + 2, 2) + 4 %e 2 5 %pi (x + 1) (a6 sin(3 %pi (x + 1)) + a5 sin(-------------) + a4 sin(2 %pi (x + 1)) 2 3 %pi (x + 1) %pi (x + 1) + a3 sin(-------------) + a2 sin(%pi (x + 1)) + a1 sin(-----------) + x + 2) 2 2 5 %pi (x + 1) 5 %pi a5 cos(-------------) 2 + expt(3 %pi a6 cos(3 %pi (x + 1)) + --------------------------- 2 3 %pi (x + 1) 3 %pi a3 cos(-------------) 2 + 2 %pi a4 cos(2 %pi (x + 1)) + --------------------------- 2 %pi (x + 1) %pi a1 cos(-----------) 2 + %pi a2 cos(%pi (x + 1)) + ----------------------- + 1, 2) - x sin(x) 2 (%i17) iyr:integrate(Fyr,x,-1,1); 15 13 11 (%o17) ((4374000 %e %pi + 26579340 %e %pi + 42899220 %e %pi 9 7 5 3 + 28459860 %e %pi + 9154860 %e %pi + 1506960 %e %pi + 121920 %e %pi 2 2 12 + 3840 %e %pi) a6 + ((- 648000 %e - 648000 %e + 648000) %pi 2 10 + (- 3793680 %e - 3865680 %e + 3793680) %pi 2 8 + (- 5504400 %e - 5925920 %e + 5504400) %pi 2 6 + (- 2946240 %e - 3557840 %e + 2946240) %pi 2 4 + (- 633600 %e - 960960 %e + 633600) %pi 2 2 + (- 46080 %e - 116480 %e + 46080) %pi - 5120 %e) a6 15 13 11 9 + (3037500 %e %pi + 18606375 %e %pi + 30677010 %e %pi + 21121815 %e %pi 7 5 3 2 + 7172880 %e %pi + 1266720 %e %pi + 111360 %e %pi + 3840 %e %pi) a5 2 12 + ((777600 %e + 1555200 %e + 777600) %pi 2 10 + (4514400 %e + 9277632 %e + 4514400) %pi 2 8 + (6388800 %e + 14222208 %e + 6388800) %pi 2 6 + (3247200 %e + 8538816 %e + 3247200) %pi 2 4 + (633600 %e + 2306304 %e + 633600) %pi 2 2 + (38400 %e + 279552 %e + 38400) %pi + 12288 %e) a5 15 13 11 9 + (1944000 %e %pi + 12083040 %e %pi + 20677020 %e %pi + 15117960 %e %pi 7 5 3 2 + 5551260 %e %pi + 1070160 %e %pi + 102720 %e %pi + 3840 %e %pi) a4 2 12 + ((- 972000 %e - 972000 %e + 972000) %pi 2 10 + (- 5555520 %e - 5798520 %e + 5555520) %pi 2 8 + (- 7500000 %e - 8888880 %e + 7500000) %pi 2 6 + (- 3461760 %e - 5336760 %e + 3461760) %pi 2 4 + (- 576000 %e - 1441440 %e + 576000) %pi 2 2 + (- 30720 %e - 174720 %e + 30720) %pi - 7680 %e) a4 15 13 11 9 + (1093500 %e %pi + 7009335 %e %pi + 12899250 %e %pi + 10448295 %e %pi 7 5 3 2 + 4290000 %e %pi + 917280 %e %pi + 96000 %e %pi + 3840 %e %pi) a3 2 12 + ((1296000 %e + 2592000 %e + 1296000) %pi 2 10 + (7155360 %e + 15462720 %e + 7155360) %pi 2 8 + (8671680 %e + 23703680 %e + 8671680) %pi 2 6 + (3261600 %e + 14231360 %e + 3261600) %pi 2 4 + (472320 %e + 3843840 %e + 472320) %pi 2 2 + (23040 %e + 465920 %e + 23040) %pi + 20480 %e) a3 15 13 11 9 + (486000 %e %pi + 3385260 %e %pi + 7343700 %e %pi + 7112820 %e %pi 7 5 3 2 + 3389100 %e %pi + 808080 %e %pi + 91200 %e %pi + 3840 %e %pi) a2 2 12 + ((- 1944000 %e - 1944000 %e + 1944000) %pi 2 10 + (- 9653040 %e - 11597040 %e + 9653040) %pi 2 8 + (- 8124720 %e - 17777760 %e + 8124720) %pi 2 6 + (- 2548800 %e - 10673520 %e + 2548800) %pi 2 4 + (- 334080 %e - 2882880 %e + 334080) %pi 2 2 + (- 15360 %e - 349440 %e + 15360) %pi - 15360 %e) a2 15 13 11 9 + (121500 %e %pi + 1210815 %e %pi + 4010370 %e %pi + 5111535 %e %pi 7 5 3 2 + 2848560 %e %pi + 742560 %e %pi + 88320 %e %pi + 3840 %e %pi) a1 2 12 + ((3888000 %e + 7776000 %e + 3888000) %pi 2 10 + (7642080 %e + 46388160 %e + 7642080) %pi 2 8 + (4987200 %e + 71111040 %e + 4987200) %pi 2 6 + (1398240 %e + 42694080 %e + 1398240) %pi 2 4 + (172800 %e + 11531520 %e + 172800) %pi 2 2 + (7680 %e + 1397760 %e + 7680) %pi + 61440 %e) a1 13 11 9 7 + (- 972000 %e %pi - 5798520 %e %pi - 8888880 %e %pi - 5336760 %e %pi 5 3 - 1441440 %e %pi - 174720 %e %pi - 7680 %e %pi) sin(1) 13 11 9 7 + (972000 %e %pi + 5798520 %e %pi + 8888880 %e %pi + 5336760 %e %pi 5 3 + 1441440 %e %pi + 174720 %e %pi + 7680 %e %pi) cos(1) 2 13 2 11 + (3888000 %e + 5184000 %e) %pi + (23194080 %e + 30925440 %e) %pi 2 9 2 7 + (35555520 %e + 47407360 %e) %pi + (21347040 %e + 28462720 %e) %pi 2 5 2 3 + (5765760 %e + 7687680 %e) %pi + (698880 %e + 931840 %e) %pi 2 13 11 + (30720 %e + 40960 %e) %pi)/(486000 %e %pi + 2899260 %e %pi 9 7 5 3 + 4444440 %e %pi + 2668380 %e %pi + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) (%i18) eq1:diff(iyr,a1)=0; 15 13 11 (%o18) (2 (121500 %e %pi + 1210815 %e %pi + 4010370 %e %pi 9 7 5 3 + 5111535 %e %pi + 2848560 %e %pi + 742560 %e %pi + 88320 %e %pi 2 12 + 3840 %e %pi) a1 + (3888000 %e + 7776000 %e + 3888000) %pi 2 10 + (7642080 %e + 46388160 %e + 7642080) %pi 2 8 + (4987200 %e + 71111040 %e + 4987200) %pi 2 6 + (1398240 %e + 42694080 %e + 1398240) %pi 2 4 + (172800 %e + 11531520 %e + 172800) %pi 2 2 + (7680 %e + 1397760 %e + 7680) %pi + 61440 %e) 13 11 9 7 /(486000 %e %pi + 2899260 %e %pi + 4444440 %e %pi + 2668380 %e %pi 5 3 + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) = 0 (%i19) eq2:diff(iyr,a2)=0; 15 13 11 (%o19) (2 (486000 %e %pi + 3385260 %e %pi + 7343700 %e %pi 9 7 5 3 + 7112820 %e %pi + 3389100 %e %pi + 808080 %e %pi + 91200 %e %pi 2 12 + 3840 %e %pi) a2 + (- 1944000 %e - 1944000 %e + 1944000) %pi 2 10 + (- 9653040 %e - 11597040 %e + 9653040) %pi 2 8 + (- 8124720 %e - 17777760 %e + 8124720) %pi 2 6 + (- 2548800 %e - 10673520 %e + 2548800) %pi 2 4 + (- 334080 %e - 2882880 %e + 334080) %pi 2 2 + (- 15360 %e - 349440 %e + 15360) %pi - 15360 %e) 13 11 9 7 /(486000 %e %pi + 2899260 %e %pi + 4444440 %e %pi + 2668380 %e %pi 5 3 + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) = 0 (%i20) eq3:diff(iyr,a3)=0; 15 13 11 (%o20) (2 (1093500 %e %pi + 7009335 %e %pi + 12899250 %e %pi 9 7 5 3 + 10448295 %e %pi + 4290000 %e %pi + 917280 %e %pi + 96000 %e %pi 2 12 + 3840 %e %pi) a3 + (1296000 %e + 2592000 %e + 1296000) %pi 2 10 + (7155360 %e + 15462720 %e + 7155360) %pi 2 8 + (8671680 %e + 23703680 %e + 8671680) %pi 2 6 + (3261600 %e + 14231360 %e + 3261600) %pi 2 4 + (472320 %e + 3843840 %e + 472320) %pi 2 2 + (23040 %e + 465920 %e + 23040) %pi + 20480 %e) 13 11 9 7 /(486000 %e %pi + 2899260 %e %pi + 4444440 %e %pi + 2668380 %e %pi 5 3 + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) = 0 (%i21) eq4:diff(iyr,a4)=0; 15 13 11 (%o21) (2 (1944000 %e %pi + 12083040 %e %pi + 20677020 %e %pi 9 7 5 3 + 15117960 %e %pi + 5551260 %e %pi + 1070160 %e %pi + 102720 %e %pi 2 12 + 3840 %e %pi) a4 + (- 972000 %e - 972000 %e + 972000) %pi 2 10 + (- 5555520 %e - 5798520 %e + 5555520) %pi 2 8 + (- 7500000 %e - 8888880 %e + 7500000) %pi 2 6 + (- 3461760 %e - 5336760 %e + 3461760) %pi 2 4 + (- 576000 %e - 1441440 %e + 576000) %pi 2 2 + (- 30720 %e - 174720 %e + 30720) %pi - 7680 %e) 13 11 9 7 /(486000 %e %pi + 2899260 %e %pi + 4444440 %e %pi + 2668380 %e %pi 5 3 + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) = 0 (%i22) eq5:diff(iyr,a5)=0; 15 13 11 (%o22) (2 (3037500 %e %pi + 18606375 %e %pi + 30677010 %e %pi 9 7 5 3 + 21121815 %e %pi + 7172880 %e %pi + 1266720 %e %pi + 111360 %e %pi 2 12 + 3840 %e %pi) a5 + (777600 %e + 1555200 %e + 777600) %pi 2 10 + (4514400 %e + 9277632 %e + 4514400) %pi 2 8 + (6388800 %e + 14222208 %e + 6388800) %pi 2 6 + (3247200 %e + 8538816 %e + 3247200) %pi 2 4 + (633600 %e + 2306304 %e + 633600) %pi 2 2 + (38400 %e + 279552 %e + 38400) %pi + 12288 %e) 13 11 9 7 /(486000 %e %pi + 2899260 %e %pi + 4444440 %e %pi + 2668380 %e %pi 5 3 + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) = 0 (%i23) eq6:diff(iyr,a6)=0; 15 13 11 (%o23) (2 (4374000 %e %pi + 26579340 %e %pi + 42899220 %e %pi 9 7 5 3 + 28459860 %e %pi + 9154860 %e %pi + 1506960 %e %pi + 121920 %e %pi 2 12 + 3840 %e %pi) a6 + (- 648000 %e - 648000 %e + 648000) %pi 2 10 + (- 3793680 %e - 3865680 %e + 3793680) %pi 2 8 + (- 5504400 %e - 5925920 %e + 5504400) %pi 2 6 + (- 2946240 %e - 3557840 %e + 2946240) %pi 2 4 + (- 633600 %e - 960960 %e + 633600) %pi 2 2 + (- 46080 %e - 116480 %e + 46080) %pi - 5120 %e) 13 11 9 7 /(486000 %e %pi + 2899260 %e %pi + 4444440 %e %pi + 2668380 %e %pi 5 3 + 720720 %e %pi + 87360 %e %pi + 3840 %e %pi) = 0 (%i24) resh:solve([eq1,eq2,eq3,eq4,eq5,eq6],[a1,a2,a3,a4,a5,a6]); 2 2 (16 %e + 32 %e + 16) %pi + 128 %e (%o24) [[a1 = - -----------------------------------, 5 3 %e %pi + 8 %e %pi + 16 %e %pi 2 2 (2 %e + 2 %e - 2) %pi + 2 %e a2 = ------------------------------, a3 = 5 3 %e %pi + 2 %e %pi + %e %pi 2 2 (144 %e + 288 %e + 144) %pi + 128 %e - --------------------------------------, 5 3 243 %e %pi + 216 %e %pi + 48 %e %pi 2 2 (4 %e + 4 %e - 4) %pi + %e a4 = -------------------------------, a5 = 5 3 16 %e %pi + 8 %e %pi + %e %pi 2 2 (400 %e + 800 %e + 400) %pi + 128 %e - ---------------------------------------, 5 3 3125 %e %pi + 1000 %e %pi + 80 %e %pi 2 2 (18 %e + 18 %e - 18) %pi + 2 %e a6 = -----------------------------------]] 5 3 243 %e %pi + 54 %e %pi + 3 %e %pi (%i25) s:subst(resh,yr); 2 2 ((18 %e + 18 %e - 18) %pi + 2 %e) sin(3 %pi (x + 1)) (%o25) ------------------------------------------------------ 5 3 243 %e %pi + 54 %e %pi + 3 %e %pi 2 2 5 %pi (x + 1) ((400 %e + 800 %e + 400) %pi + 128 %e) sin(-------------) 2 - ----------------------------------------------------------- 5 3 3125 %e %pi + 1000 %e %pi + 80 %e %pi 2 2 ((4 %e + 4 %e - 4) %pi + %e) sin(2 %pi (x + 1)) + ------------------------------------------------- 5 3 16 %e %pi + 8 %e %pi + %e %pi 2 2 3 %pi (x + 1) ((144 %e + 288 %e + 144) %pi + 128 %e) sin(-------------) 2 - ----------------------------------------------------------- 5 3 243 %e %pi + 216 %e %pi + 48 %e %pi 2 2 ((2 %e + 2 %e - 2) %pi + 2 %e) sin(%pi (x + 1)) + ------------------------------------------------- 5 3 %e %pi + 2 %e %pi + %e %pi 2 2 %pi (x + 1) ((16 %e + 32 %e + 16) %pi + 128 %e) sin(-----------) 2 - ------------------------------------------------------ + x + 2 5 3 %e %pi + 8 %e %pi + 16 %e %pi (%i26) plot2d([s, rhs(extr)],[x, -1, 1]); plot
![plot2d([s, rhs(extr)],[x, -1, 1]);](http://maxima-online.org/plot.html?g=p-446290715.png&t=img&db=r-1056006533)
Run Example
eqn:(Dy)^2+y^2+4*y*exp(x)-x*sin(x); 2 x 2 (%o1) y + 4 %e y - x sin(x) + Dy (%i2) h:diff(eqn,y); x (%o2) 2 y + 4 %e (%i3) h1:diff(eqn,Dy); (%o3) 2 Dy (%i4) euler:h-(diff(h1, x) + diff(h1, y)*Dy+diff(h1, Dy)*'diff(Dy, x))=0; x dDy (%o4) 2 y + 4 %e - 2 --- = 0 dx (%i5) eqn1:subst(Dy='diff(y, x), euler); 2 d y x (%o5) - 2 --- + 2 y + 4 %e = 0 2 dx (%i6) sol:ode2(eqn1, y, x); x (2 x - 1) %e x - x (%o6) y = ------------- + %k1 %e + %k2 %e 2 (%i7) extr:bc2(sol, x=-1, y=1, x=1, y=3); x 3 2 x 2 - x (2 x - 1) %e (%e - 5 %e - 5 %e - 3) %e (%e + 3 %e) %e (%o7) y = ------------- - ---------------------------- + ------------------ 2 3 2 3 2 2 %e + 2 %e + 2 %e + 2 %e + %e + %e + 1 (%i8) x1:-1; (%o8) - 1 (%i9) y1:1; (%o9) 1 (%i10) x2:1; (%o10) 1 (%i11) y2:3; (%o11) 3 (%i12) I:y; (%o12) y (%i13) L:eqn+%l*y; 2 x 2 (%o13) y + 4 %e y + %l y - x sin(x) + Dy (%i14) lh:diff(L,y); x (%o14) 2 y + 4 %e + %l (%i15) lh1:diff(L,Dy); (%o15) 2 Dy (%i16) eulerL:lh-(diff(lh1, x) + diff(lh1, y)*Dy+diff(lh1, Dy)*'diff(Dy, x))=-3; x dDy (%o16) 2 y + 4 %e - 2 --- + %l = - 3 dx (%i17) eqn2:subst(Dy='diff(y, x), eulerL); 2 d y x (%o17) - 2 --- + 2 y + 4 %e + %l = - 3 2 dx (%i18) sol1:ode2(eqn2, y, x); x (2 x - 1) %e - %l - 3 x - x (%o18) y = ---------------------- + %k1 %e + %k2 %e 2 (%i19) ur1:subst([x=x1,y=y1],sol1); - 1 - %l - 3 %e - 3 - 1 (%o19) 1 = ------------------ + %e %k2 + %e %k1 2 (%i20) ur2:subst([x=x2,y=y2],sol1); - %l + %e - 3 - 1 (%o20) 3 = ------------- + %e %k2 + %e %k1 2 (%i21) Dsol1:diff(sol1,x); x x (2 x - 1) %e + 2 %e x - x (%o21) 0 = --------------------- + %k1 %e - %k2 %e 2 (%i22) ir:integrate(subst(y=rhs(sol1),I),x,x1,x2)=1.5; (%o22) - 1 2 2 2 %e (2 %e %l + (2 - 2 %e ) %k2 + (2 - 2 %e ) %k1 + %e + 6 %e - 5) - -------------------------------------------------------------------- = 1.5 2 (%i23) solve1:solve([ur1,ur2,ir],[%k1,%k2,%l]); rat: replaced -1.5 by -3/2 = -1.5 3 2 2 %e - 3 %e - 11 %e - 14 (%o23) [[%k1 = - --------------------------, 8 %e + 8 3 2 2 %e - 3 %e - 15 %e - 2 %k2 = - -------------------------, %l = 8 %e + 8 - 1 4 3 2 %e (2 %e - 5 %e - 8 %e + 23 %e - 2) - -----------------------------------------]] 4 (%i24) resh:subst(solve1,sol1); - 1 4 3 2 x %e (2 %e - 5 %e - 8 %e + 23 %e - 2) (2 x - 1) %e + ----------------------------------------- - 3 4 (%o24) y = ------------------------------------------------------------- 2 3 2 x 3 2 - x (2 %e - 3 %e - 11 %e - 14) %e (2 %e - 3 %e - 15 %e - 2) %e - -------------------------------- - --------------------------------- 8 %e + 8 8 %e + 8 (%i25) plot2d([rhs(resh), rhs(extr)],[x, -1, 1]); plot
![plot2d([rhs(resh), rhs(extr)],[x, -1, 1]);](http://maxima-online.org/plot.html?g=p-1861722578.png&t=img&db=r-1110920433)
Run Example
euler (50); (%o1) - 6053285248188621896314383785111649088103498225146815121 (%i2)
[atan,diff,euler,is,ratsimp,sin,trigexpand] [bc2,cos,diff,euler,integrate,ode2,plot2d,rhs,solve,subst] [bc2,diff,euler,exp,integrate,ode2,plot2d,rhs,sin,subst] [cos,diff,euler,factor,is,ratsimp] [determinant,euler,ev,matrix,mod,totient] [determinant,euler,matrix,mod,totient] [diff,euler,factor,is,ratsimp] [diff,euler,factorial,is,ratsimp] [euler]
