Kitonum

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These are answers submitted by Kitonum

Use  fsolve  with  initialpoint  option :

fsolve([Asubs/Bsubs = 1, 2*Esubs = h*`J__ω`[49]], {a = 2*10^(-20) .. 4*10^(-20), k = 4*10^8 .. 8*10^8});

                            {a = 3.31129407052*10^(-20), k = 6.47179093038*10^8}


I got this this initialpoint  {a = 2*10^(-20) .. 4*10^(-20), k = 4*10^8 .. 8*10^8}  from the implicitplot. If you want to automate this process for a large number of applications, then try using the  DirectSearch package, which you can freely download from Maple Application Center.

Here is a procedure that does this not only for lists, but also for expressions similar to polynomials. Formal parameters: P - a list or an expression of type `+`,  var is a list of symbols for which the procedure isolates nonlinear members.

IsolateNonlinearTerms:=proc(P::{list,`+`,polynom},var::list)
uses ListTools;
selectremove(p->degree(p)>1 or degree(p)<0, `if`(P::list,FlattenOnce(P),[op(P)]),var);
end proc:


Examples of use:

w:=[[z, y, x, x^3, 1, -5*y], [x*z, x*y, y, 2*x, 1], [x*z, 1/z^2, z, x*y]];
IsolateNonlinearTerms(w, [x,y,z]);
IsolateNonlinearTerms(x^3-3*x*y^2+y^4+x-5*y+4*z-10, [x,y]);
                         

 

 

restart;
pde1 := (y+z)*(diff(u(x, y, z), x))+(z+x)*(diff(u(x, y, z), y))+(x+y)*(diff(u(x, y, z), z)) = 0;
pdsolve(pde1, u(x,y,z), HINT=strip);


See help on  pdsolve  command  for details.


 

restart:  with(plots):

cnsts := [ 4*x1 + x2 <= 12,
             x1 - x2 >= 2,
                  x1 >= 0,
                  x2 >= 0 ];

[4*x1+x2 <= 12, 2 <= x1-x2, 0 <= x1, 0 <= x2]

(1)

feasibleRegion := inequal(cnsts, x1 = 0 .. 4, x2 = -1 .. 2, nolines):
display(feasibleRegion);

 

animate(implicitplot, [c1*x1+2*x2=0,x1=0..4,x2=-1..2, color=red],c1 = -1 .. 0,
                frames = 50, background = feasibleRegion);

 

 


Edit. Below are 2 more options with animation. In the first case, we plot the function of 2 variables for each value of c1. The third coordinate of the highest point gives you the desired maximum. In the second option, we directly plot the maximum as a function of c1:


 

restart:  with(plots):

cnsts := [ 4*x1 + x2 <= 12,
             x1 - x2 >= 2,
                  x1 >= 0,
                  x2 >= 0 ];

[4*x1+x2 <= 12, 2 <= x1-x2, 0 <= x1, 0 <= x2]

(1)

feasibleRegion := inequal(cnsts, x1 = 2 .. 3, x2 = 0 .. 1, nolines):
display(feasibleRegion);
FeasibleRegion:=display(plottools:-transform((x,y)->[x,y,0])(feasibleRegion));
solve({4*x1 + x2 = 12, x1 - x2 = 2});

 

 

{x1 = 14/5, x2 = 4/5}

(2)

animate(plot3d, [c1*x1+2*x2,x1=x2+2..(12-x2)/4,x2=0..4/5, style=surface, color=khaki],c1 = -10 .. 100, frames=90, background=FeasibleRegion);
animate(plot, [[t,'Optimization:-Maximize'(t*x1+2*x2,cnsts)[1],t=-10..c1]],c1 = -10 .. 100, frames=60);

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

Warning, problem appears to be unbounded

 

 

 


 

Download question_on_animate_with_background_new2.mw

Download question_on_animate_with_background_new1.mw


 

restart;
with(plots):
F:=2+r*(b-2)*sqrt(b-1):
A:=implicitplot([b=2,F], b=-1..5, r=-5..10, linestyle=[3,1],color=[black,red], thickness=[0,2], gridrefine=3):
B:=inequal(F<0, b=1..5, r=-5..10, color="LightBlue", optionsexcluded = [color = "Pink"],nolines, numpoints=10000):

T:=textplot([[1.7,7,"F=0"],[2.5,-3,"F=0"],[4,-3,"F<0"],[1.4,9,"F<0"],[3,3,"F>0"]], font=[times,bold,16]):
display(A,B,T, size=[500,500], view=[-1..5,-5..10]);

 

# The red curve  F=0 can also be plotted explicitly 
plot(-2/(b-2)/sqrt(b-1), b=-1..5, -5..10, color=red, thickness=2, discont);

 

 


 

Download plot5.mw

Another way to find the roots of the equation in a certain range is  RootFinding:-Analytic  command. It works numerically, but the use of  identify  command in some cases allows you to get symbolic (i.e. exact) root values. For comparison, the same example was solved by  Student:-Calculus1:-Roots  command (2000 times slower).

 

restart;
Eq:=sin(x)=simplify(convert(sin(Pi/60),radical));

sin(x) = (1/16)*(-2*3^(1/2)+2)*(5+5^(1/2))^(1/2)+(1/16)*2^(1/2)*(5^(1/2)-1)*(1+3^(1/2))

(1)

CodeTools:-Usage(RootFinding:-Analytic(Eq, re=3*Pi/2..5*Pi, im=-1..1));
evalf[12]([%]);
identify(%);

memory used=3.55MiB, alloc change=0 bytes, cpu time=46.00ms, real time=82.00ms, gc time=0ns

 

9.37241808320955, 15.6556033903891, 12.6187304919190, 6.33554518473960

 

[9.37241808321, 15.6556033904, 12.6187304919, 6.33554518474]

 

[(179/60)*Pi, (299/60)*Pi, (241/60)*Pi, (121/60)*Pi]

(2)

ts:=time():
Student:-Calculus1:-Roots(Eq, x=3*Pi/2..5*Pi);
time()-ts;
identify(%%);

Warning, some roots are returned as numeric approximations

 

[6.335545185, 9.372418083, 12.61873049, 15.65560339]

 

187.782

 

[(121/60)*Pi, (179/60)*Pi, (241/60)*Pi, (299/60)*Pi]

(3)

187.782/0.082;

2290.024390

(4)

 


 

Download roots.mw

It follows from the obvious symmetry of your equations that any pair of equal numbers  xa1=xb1   is a solution to this system. In addition, if we plot graphs, then it is clearly visible on them that there are a couple of solutions when the roots are not equal. One of these pairs was indicated by Carl in his reply. Another symmetric solution is found below:


 

restart;

T:=325:

a12:=2305.28444347652 - 9.14490843016421*T + 0.00680052257590234*T^2:

a21:=-6665.24838284836 + 46.0897018087247*T - 0.0694991633494123*T^2:

alfa:=0.3:

x2:=1-x1:

tau12:=a12/T:

tau21:=a21/T:

G12:=exp(-alfa*tau12):

G21:=exp(-alfa*tau21):

lng1:=x2^2*(tau21*(G21/(x1+x2*G21))^2+tau12*(G12/((x2+x1*G12)^2))):

lng2:=x1^2*(tau12*(G12/(x2+x1*G12))^2+tau21*(G21/((x1+x2*G21)^2))):

lnga1:=subs(x1=xa1,lng1):

lngb1:=subs(x1=xb1,lng1):

lnga2:=subs(x1=xa1,lng2):

lngb2:=subs(x1=xb1,lng2):

r1:=lnga1+ln(xa1)=lngb1+ln(xb1);

(1-xa1)^2*(.4966874722/(.4073000470+.5926999530*xa1)^2+.1510891213/(-0.464212743e-1*xa1+1)^2)+ln(xa1) = (1-xb1)^2*(.4966874722/(.4073000470+.5926999530*xb1)^2+.1510891213/(-0.464212743e-1*xb1+1)^2)+ln(xb1)

(1)

r2:=lnga2+ln(1-xa1)=lngb2+ln(1-xb1);

xa1^2*(.1440753718/(-0.464212743e-1*xa1+1)^2+1.219463331/(.4073000470+.5926999530*xa1)^2)+ln(1-xa1) = xb1^2*(.1440753718/(-0.464212743e-1*xb1+1)^2+1.219463331/(.4073000470+.5926999530*xb1)^2)+ln(1-xb1)

(2)

A:=plots:-implicitplot(r1, xa1=0..1, xb1=0..1, color=red, gridrefine=4);
B:=plots:-implicitplot(r2, xa1=0..1, xb1=0..1, color=blue, gridrefine=4);
plots:-display(A,B);

# One of the solutions at the top
fsolve({r1,r2},{xa1=0..0.2,xb1=0.4..0.6});

 

 

 

{xa1 = 0.6850883394e-1, xb1 = .5048762298}

(3)

 


 

Download NRTL_new.mw

 

Maybe a typo has occurred? If we remove the minus in front of  a , and increase the coefficient itself, we get a very similar graph:

restart;
plots:-implicitplot((7.72-7.72*B)*(25.717267500*a) = 662204.4444*B^2, a = 10 .. 50000, B = 0.01 .. 1, color="Blue", thickness=3, tickmarks=[7,10], gridrefine=3, size=[700,400], gridlines, view=[0..60000,0..1]);

             

 

restart;
P:=x->x^4+x^3+a*x^2+sqrt(2)*x+b;
solve(evalc([Re,Im](P(1+I)))=~[0,0]);
solve(eval(P(x),%));

                

 

 

This object  sq1  is not a square. Therefore, we cannot interpret this as a bug. The error is rather that Maple does not test it for the correctness of your definition. Maple considers its area as if it is in fact a square, that is, half of the product of the diagonals. Hence 2.5


 

restart

Eq1 := (2.394038482*10^(-25)*A[1]*B[1]*b[1]*ln(4624/3969)*a[1]^2 + 6.231123984*10^(-25)*a[1]^2*b[1]*ln(4624/3969)^2*A[1]*B[1] + 8.857755670*10^(-26)*a[1]^3*ln(4624/3969)^3*B[1]^2 + 1.856626218*10^(-33)*a[1]*ln(4624/3969)^2*A[1]^2 + 3.115561992*10^(-25)*a[1]*b[1]^2*ln(4624/3969)^2*A[1]^2 + 2.657326700*10^(-25)*a[1]*b[1]^2*ln(4624/3969)^3*A[1]^2 + 2.657326700*10^(-25)*a[1]*b[1]^2*ln(4624/3969)^3*B[1]^2 + 4.995877205*10^(-27)*a[1]^3*B[1]^2 + 4.023466006*10^(-35)*a[1]*B[1]^2 + 2.497938606*10^(-26)*a[1]^3*A[1]^2 + 5.314653400*10^(-25)*a[1]^2*b[1]*ln(4624/3969)^3*A[1]*B[1] + 1.995032068*10^(-25)*A[1]^2*ln(4624/3969)*a[1]^3 + 4.428877833*10^(-25)*a[1]^3*ln(4624/3969)^3*A[1]^2 + 5.192603320*10^(-25)*a[1]^3*ln(4624/3969)^2*A[1]^2 + 1.038520664*10^(-25)*a[1]^3*ln(4624/3969)^2*B[1]^2 + 1.498763163*10^(-26)*a[1]*b[1]^2*A[1]^2 + 1.498763163*10^(-26)*a[1]*b[1]^2*B[1]^2 + 8.199429997*10^(-34)*a[1]*ln(4624/3969)*A[1]^2 + 4.671138947*10^(-34)*a[1]*ln(4624/3969)^3*B[1]^2 + 1.401341684*10^(-33)*a[1]*ln(4624/3969)^3*A[1]^2 + 3.990064137*10^(-26)*B[1]^2*ln(4624/3969)*a[1]^3 + 9.991754410*10^(-27)*A[1]*B[1]*b[1]^3 + 8.046932010*10^(-35)*A[1]*B[1]*b[1] + 3.115561992*10^(-25)*a[1]*b[1]^2*ln(4624/3969)^2*B[1]^2 + 2.997526324*10^(-26)*a[1]^2*b[1]*A[1]*B[1] + 1.237750812*10^(-33)*A[1]*B[1]*b[1]*ln(4624/3969)^2 + 6.188754060*10^(-34)*a[1]*ln(4624/3969)^2*B[1]^2 + 2.077041328*10^(-25)*b[1]^3*ln(4624/3969)^2*A[1]*B[1] + 1.771551133*10^(-25)*b[1]^3*ln(4624/3969)^3*A[1]*B[1] + 7.980128275*10^(-26)*A[1]*B[1]*b[1]^3*ln(4624/3969) + 5.466286665*10^(-34)*A[1]*B[1]*b[1]*ln(4624/3969) + 9.342277895*10^(-34)*A[1]*B[1]*b[1]*ln(4624/3969)^3 - 8.980366659*10^(-50)*b[1]*ln(4624/3969)^5 + 8.628745640*10^(-49)*a[1]*ln(4624/3969)^4 - 1.983002476*10^(-49)*b[1]*ln(4624/3969)^4 + 3.907675385*10^(-49)*a[1]*ln(4624/3969)^5 + 1.207039802*10^(-34)*a[1]*A[1]^2 + 1.197019241*10^(-25)*A[1]^2*b[1]^2*ln(4624/3969)*a[1] + 1.197019241*10^(-25)*B[1]^2*b[1]^2*ln(4624/3969)*a[1] + 2.733143333*10^(-34)*a[1]*ln(4624/3969)*B[1]^2 - 1.751509252*10^(-49)*b[1]*ln(4624/3969)^3 + 3.365859858*10^(-49)*a[1]*ln(4624/3969)^2 + 7.621436685*10^(-49)*a[1]*ln(4624/3969)^3 - 1.708050894*10^(-50)*b[1]*ln(4624/3969) - 7.735201281*10^(-50)*b[1]*ln(4624/3969)^2 + 7.432333988*10^(-50)*ln(4624/3969)*a[1] - 1.508655173*10^(-51)*b[1] + 6.564692631*10^(-51)*a[1])/(4.097832766*10^(-51)*ln(4624/3969)^5 + 9.048642256*10^(-51)*ln(4624/3969)^4 + 7.992315096*10^(-51)*ln(4624/3969)^3 + 3.529651123*10^(-51)*ln(4624/3969)^2 + 7.794010183*10^(-52)*ln(4624/3969) + 6.884147200*10^(-53)):

Eq2 := (6.188754060*10^(-34)*b[1]*A[1]^2*ln(4624/3969)^2 + 9.991754410*10^(-27)*a[1]^3*A[1]*B[1] + 8.199429997*10^(-34)*b[1]*B[1]^2*ln(4624/3969) + 1.498763163*10^(-26)*a[1]^2*b[1]*A[1]^2 + 1.401341684*10^(-33)*b[1]*B[1]^2*ln(4624/3969)^3 + 1.197019241*10^(-25)*b[1]*B[1]^2*ln(4624/3969)*a[1]^2 + 1.197019241*10^(-25)*b[1]*A[1]^2*ln(4624/3969)*a[1]^2 + 2.497938606*10^(-26)*B[1]^2*b[1]^3 - 1.594466862*10^(-55)*ln(4624/3969)^3 - 7.041653990*10^(-56)*ln(4624/3969)^2 + 2.394038482*10^(-25)*A[1]*B[1]*b[1]^2*ln(4624/3969)*a[1] + 1.038520664*10^(-25)*b[1]^3*ln(4624/3969)^2*A[1]^2 + 5.192603320*10^(-25)*b[1]^3*ln(4624/3969)^2*B[1]^2 + 8.980366659*10^(-50)*a[1]*ln(4624/3969)^5 + 2.657326700*10^(-25)*a[1]^2*b[1]*ln(4624/3969)^3*B[1]^2 + 4.023466006*10^(-35)*b[1]*A[1]^2 + 1.207039802*10^(-34)*b[1]*B[1]^2 + 4.671138947*10^(-34)*b[1]*A[1]^2*ln(4624/3969)^3 + 1.856626218*10^(-33)*b[1]*B[1]^2*ln(4624/3969)^2 - 8.175176368*10^(-56)*ln(4624/3969)^5 - 1.805204130*10^(-55)*ln(4624/3969)^4 + 3.115561992*10^(-25)*a[1]^2*b[1]*ln(4624/3969)^2*A[1]^2 + 3.115561992*10^(-25)*a[1]^2*b[1]*ln(4624/3969)^2*B[1]^2 + 2.657326700*10^(-25)*a[1]^2*b[1]*ln(4624/3969)^3*A[1]^2 + 5.314653400*10^(-25)*a[1]*b[1]^2*ln(4624/3969)^3*A[1]*B[1] + 6.231123984*10^(-25)*a[1]*b[1]^2*ln(4624/3969)^2*A[1]*B[1] - 1.373387366*10^(-57) + 3.990064137*10^(-26)*b[1]^3*A[1]^2*ln(4624/3969) + 3.365859858*10^(-49)*b[1]*ln(4624/3969)^2 + 7.735201281*10^(-50)*a[1]*ln(4624/3969)^2 + 1.498763163*10^(-26)*a[1]^2*b[1]*B[1]^2 + 2.733143333*10^(-34)*b[1]*A[1]^2*ln(4624/3969) + 1.237750812*10^(-33)*a[1]*ln(4624/3969)^2*A[1]*B[1] + 5.466286665*10^(-34)*A[1]*B[1]*ln(4624/3969)*a[1] + 9.342277895*10^(-34)*a[1]*ln(4624/3969)^3*A[1]*B[1] - 1.554905032*10^(-56)*ln(4624/3969) + 8.857755670*10^(-26)*b[1]^3*ln(4624/3969)^3*A[1]^2 + 1.995032068*10^(-25)*B[1]^2*b[1]^3*ln(4624/3969) + 8.046932010*10^(-35)*A[1]*B[1]*a[1] + 4.428877833*10^(-25)*b[1]^3*ln(4624/3969)^3*B[1]^2 + 2.077041328*10^(-25)*a[1]^3*ln(4624/3969)^2*A[1]*B[1] + 2.997526324*10^(-26)*a[1]*b[1]^2*A[1]*B[1] + 7.980128275*10^(-26)*A[1]*B[1]*ln(4624/3969)*a[1]^3 + 1.771551133*10^(-25)*a[1]^3*ln(4624/3969)^3*A[1]*B[1] + 1.983002476*10^(-49)*a[1]*ln(4624/3969)^4 + 8.628745640*10^(-49)*b[1]*ln(4624/3969)^4 + 3.907675385*10^(-49)*b[1]*ln(4624/3969)^5 + 4.995877205*10^(-27)*b[1]^3*A[1]^2 + 7.621436685*10^(-49)*b[1]*ln(4624/3969)^3 + 1.751509252*10^(-49)*a[1]*ln(4624/3969)^3 + 7.432333988*10^(-50)*b[1]*ln(4624/3969) + 1.708050894*10^(-50)*ln(4624/3969)*a[1] + 6.564692631*10^(-51)*b[1] + 1.508655173*10^(-51)*a[1])/(4.097832766*10^(-51)*ln(4624/3969)^5 + 9.048642256*10^(-51)*ln(4624/3969)^4 + 7.992315096*10^(-51)*ln(4624/3969)^3 + 3.529651123*10^(-51)*ln(4624/3969)^2 + 7.794010183*10^(-52)*ln(4624/3969) + 6.884147200*10^(-53)):

Eq3 := (6.795005989*10^(-42)*a[1]^4*ln(4624/3969)^3*A[1] + 4.209850900*10^(-42)*a[1]^4*ln(4624/3969)^4*A[1] + 1.359001197*10^(-42)*b[1]^4*ln(4624/3969)^3*A[1] + 8.419701800*10^(-43)*b[1]^4*ln(4624/3969)^4*A[1] + 8.388879275*10^(-44)*a[1]*b[1]^3*B[1] - 1.228735462*10^(-57)*A[1]*ln(4624/3969)^5 + 1.074935208*10^(-42)*A[1]*ln(4624/3969)*a[1]^4 - 3.754479537*10^(-60)*B[1] - 2.064212054*10^(-59)*A[1] - 1.340926813*10^(-57)*B[1]*ln(4624/3969)^5 - 5.060514119*10^(-58)*B[1]*ln(4624/3969)^6 + 8.388879275*10^(-44)*a[1]^3*b[1]*B[1] + 6.756044870*10^(-52)*a[1]*b[1]*B[1] + 1.776052466*10^(-50)*a[1]*b[1]*ln(4624/3969)^4*B[1] + 3.367880720*10^(-42)*a[1]*b[1]^3*ln(4624/3969)^4*B[1] + 5.436004790*10^(-42)*a[1]*b[1]^3*ln(4624/3969)^3*B[1] + 2.097219818*10^(-44)*b[1]^4*A[1] + 3.367880720*10^(-42)*a[1]^3*b[1]*ln(4624/3969)^4*B[1] + 5.436004790*10^(-42)*a[1]^3*b[1]*ln(4624/3969)^3*B[1] + 5.051821080*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)^4*A[1] + 8.154007187*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)^3*A[1] + 1.289922249*10^(-42)*b[1]^2*A[1]*ln(4624/3969)*a[1]^2 + 8.599481665*10^(-43)*b[1]*B[1]*ln(4624/3969)*a[1]^3 + 8.599481665*10^(-43)*b[1]^3*B[1]*ln(4624/3969)*a[1] + 3.260938160*10^(-42)*a[1]^3*b[1]*ln(4624/3969)^2*B[1] + 4.891407240*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)^2*A[1] - 1.058366154*10^(-57)*ln(4624/3969)^2*A[1] - 2.887504260*10^(-58)*ln(4624/3969)^2*B[1] - 2.396496281*10^(-57)*A[1]*ln(4624/3969)^3 - 2.337034537*10^(-58)*A[1]*ln(4624/3969) + 3.378022435*10^(-52)*b[1]^2*A[1] + 1.048609909*10^(-43)*a[1]^4*A[1] + 1.013406730*10^(-51)*a[1]^2*A[1] - 2.713236060*10^(-57)*A[1]*ln(4624/3969)^4 - 8.717705361*10^(-58)*B[1]*ln(4624/3969)^3 - 5.100841261*10^(-59)*B[1]*ln(4624/3969) - 1.480485871*10^(-57)*B[1]*ln(4624/3969)^4 + 6.119181470*10^(-51)*b[1]*B[1]*ln(4624/3969)*a[1] + 3.137436654*10^(-50)*a[1]*b[1]*ln(4624/3969)^3*B[1] + 2.078382172*10^(-50)*a[1]*b[1]*ln(4624/3969)^2*B[1] + 3.260938160*10^(-42)*a[1]*b[1]^3*ln(4624/3969)^2*B[1] + 2.664078699*10^(-50)*a[1]^2*ln(4624/3969)^4*A[1] + 2.149870415*10^(-43)*b[1]^4*A[1]*ln(4624/3969) + 4.706154981*10^(-50)*a[1]^2*ln(4624/3969)^3*A[1] + 9.178772213*10^(-51)*A[1]*ln(4624/3969)*a[1]^2 + 3.117573259*10^(-50)*a[1]^2*ln(4624/3969)^2*A[1] + 8.880262330*10^(-51)*b[1]^2*A[1]*ln(4624/3969)^4 + 3.059590737*10^(-51)*b[1]^2*A[1]*ln(4624/3969) + 8.152345410*10^(-43)*b[1]^4*ln(4624/3969)^2*A[1] + 4.076172701*10^(-42)*a[1]^4*ln(4624/3969)^2*A[1] + 1.039191087*10^(-50)*b[1]^2*ln(4624/3969)^2*A[1] + 1.568718327*10^(-50)*b[1]^2*A[1]*ln(4624/3969)^3 + 1.258331891*10^(-43)*a[1]^2*b[1]^2*A[1])/(5.196166686*10^(-61)*ln(4624/3969)^6 + 1.376871810*10^(-60)*ln(4624/3969)^5 + 1.520171900*10^(-60)*ln(4624/3969)^4 + 8.951392907*10^(-61)*ln(4624/3969)^3 + 2.964906943*10^(-61)*ln(4624/3969)^2 + 5.237574842*10^(-62)*ln(4624/3969) + 3.855122432*10^(-63)):

Eq4 := (1.340926813*10^(-57)*A[1]*ln(4624/3969)^5 + 3.754479537*10^(-60)*A[1] + 8.717705361*10^(-58)*A[1]*ln(4624/3969)^3 - 1.228735462*10^(-57)*B[1]*ln(4624/3969)^5 + 4.891407240*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)^2*B[1] + 5.051821080*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)^4*B[1] + 8.388879275*10^(-44)*a[1]*b[1]^3*A[1] + 6.756044870*10^(-52)*a[1]*b[1]*A[1] + 8.388879275*10^(-44)*a[1]^3*b[1]*A[1] + 3.117573259*10^(-50)*b[1]^2*ln(4624/3969)^2*B[1] + 4.706154981*10^(-50)*b[1]^2*ln(4624/3969)^3*B[1] + 1.568718327*10^(-50)*a[1]^2*ln(4624/3969)^3*B[1] + 1.039191087*10^(-50)*a[1]^2*ln(4624/3969)^2*B[1] + 2.664078699*10^(-50)*b[1]^2*ln(4624/3969)^4*B[1] + 9.178772213*10^(-51)*b[1]^2*ln(4624/3969)*B[1] + 8.880262330*10^(-51)*a[1]^2*ln(4624/3969)^4*B[1] + 4.209850900*10^(-42)*b[1]^4*ln(4624/3969)^4*B[1] + 6.795005989*10^(-42)*b[1]^4*ln(4624/3969)^3*B[1] + 3.059590737*10^(-51)*a[1]^2*ln(4624/3969)*B[1] + 2.149870415*10^(-43)*ln(4624/3969)*a[1]^4*B[1] + 1.359001197*10^(-42)*ln(4624/3969)^3*a[1]^4*B[1] + 8.419701800*10^(-43)*ln(4624/3969)^4*a[1]^4*B[1] + 8.152345410*10^(-43)*ln(4624/3969)^2*a[1]^4*B[1] + 4.076172701*10^(-42)*b[1]^4*ln(4624/3969)^2*B[1] + 1.074935208*10^(-42)*b[1]^4*ln(4624/3969)*B[1] + 1.258331891*10^(-43)*a[1]^2*b[1]^2*B[1] + 3.378022435*10^(-52)*a[1]^2*B[1] + 1.013406730*10^(-51)*b[1]^2*B[1] + 5.060514119*10^(-58)*A[1]*ln(4624/3969)^6 + 2.097219818*10^(-44)*a[1]^4*B[1] + 1.048609909*10^(-43)*b[1]^4*B[1] + 1.480485871*10^(-57)*A[1]*ln(4624/3969)^4 - 2.713236060*10^(-57)*B[1]*ln(4624/3969)^4 + 2.887504260*10^(-58)*ln(4624/3969)^2*A[1] - 1.058366154*10^(-57)*ln(4624/3969)^2*B[1] - 2.396496281*10^(-57)*B[1]*ln(4624/3969)^3 + 5.100841261*10^(-59)*A[1]*ln(4624/3969) - 2.337034537*10^(-58)*B[1]*ln(4624/3969) + 1.289922249*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)*B[1] + 8.154007187*10^(-42)*a[1]^2*b[1]^2*ln(4624/3969)^3*B[1] + 3.260938160*10^(-42)*a[1]*b[1]^3*ln(4624/3969)^2*A[1] + 2.078382172*10^(-50)*a[1]*b[1]*ln(4624/3969)^2*A[1] + 3.260938160*10^(-42)*a[1]^3*b[1]*ln(4624/3969)^2*A[1] + 8.599481665*10^(-43)*a[1]^3*b[1]*ln(4624/3969)*A[1] + 8.599481665*10^(-43)*a[1]*b[1]^3*ln(4624/3969)*A[1] + 5.436004790*10^(-42)*a[1]^3*b[1]*ln(4624/3969)^3*A[1] + 1.776052466*10^(-50)*a[1]*b[1]*ln(4624/3969)^4*A[1] + 3.367880720*10^(-42)*a[1]^3*b[1]*ln(4624/3969)^4*A[1] + 5.436004790*10^(-42)*a[1]*b[1]^3*ln(4624/3969)^3*A[1] + 6.119181470*10^(-51)*a[1]*b[1]*ln(4624/3969)*A[1] + 3.137436654*10^(-50)*a[1]*b[1]*ln(4624/3969)^3*A[1] + 3.367880720*10^(-42)*a[1]*b[1]^3*ln(4624/3969)^4*A[1] - 2.064212054*10^(-59)*B[1])/(5.196166686*10^(-61)*ln(4624/3969)^6 + 1.376871810*10^(-60)*ln(4624/3969)^5 + 1.520171900*10^(-60)*ln(4624/3969)^4 + 8.951392907*10^(-61)*ln(4624/3969)^3 + 2.964906943*10^(-61)*ln(4624/3969)^2 + 5.237574842*10^(-62)*ln(4624/3969) + 3.855122432*10^(-63)):

sys := [Eq1 , Eq2, Eq3, Eq4]:
indets(sys);

{A[1], B[1], a[1], b[1]}

(1)

Digits:=40:
fsolve(sys, {a[1],b[1], A[1], B[1]});
evalf(eval(sys, %));  # Check

{A[1] = 0., B[1] = 0., a[1] = 0.4566700010633004777680982475920347128922e-7, b[1] = 0.1987132808635849788907268862388896010783e-6}

 

[-0.2437401628335635546686719686578674464878e-45, -0.3603979671175339458365827924625900164166e-44, 0., 0.]

(2)

NULL


 

Download SimulteniousEq_new.mw

restart:
FibRatios:= proc(n::posint)
option remember;
    if n=1 then return 1
    elif n=2 then return 2
    else (thisproc(n-1)*thisproc(n-2)+thisproc(n-2))/(thisproc(n-2)+1)
    fi;
end proc:

[seq(FibRatios(n), n=1..25)];
evalf(%);

 

Example:

A:=plot(x^2, x=0..3, color=red, thickness=3):
B:=plots:-arrow([3,3^2], color=blue, width=0.03, head_width=0.2, head_length=0.7):
plots:-display(A, B);

                   

See help on the  plots:-arrow command  for details.

It is very simple:

restart;
p:=t^2+2*t;
q:=t^3*(t+2);
gcd(p,q)/(t+2);


Or more generally and programmatically:

restart;
p:=t^2+2*t;
F1:=factors(p);
n1:=select(c->c[1]=t+2, F1[2])[][2]:
q:=t^3*(t+2);
F2:=factors(q);
n2:=select(c->c[1]=t+2, F2[2])[][2]:
n:=min(n1,n2);
gcd(p/(t+2)^n,q/(t+2)^n);

Here  n1  and  n2  are  the multiplicities of the root  t=-2  in polynomials  p  and  q .

Data := x(t)^2;
Physics:-diff(Data, x(t));
diff(Data, t);
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