## 14433 Reputation

11 years, 263 days

## Typo?...

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]);```

## A way...

```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),%));```

## sq1  is not a square...

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

## Digits, fsolve...

 > 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);
 (1)
 > Digits:=40: fsolve(sys, {a[1],b[1], A[1], B[1]}); evalf(eval(sys, %));  # Check
 (2)
 >

## Procedure...

```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(%);
```

## plots:-arrow...

Example:

```A:=plot(x^2, x=0..3, color=red, thickness=3):
plots:-display(A, B);```

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

## A way...

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 .

## A way...

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

## Numeric solution...

Probably your system can only be solved numerically. To do this, specify the values of all 15 parameters and set the initial conditions. I took them from 1 to 15 and arbitrarily selected 3 initial conditions (see code below):

 > restart; odeA := {m*diff(x(t), t, t) = -m*A*sin(2*Pi*f*t) - k*x(t) + 0.5*q(t)*(N__f*epsilon__0*L*(-2*x(t)/(G^2 - x(t)^2) + 2*(G__1^2 - x(t)^2)*x(t)/(G^2 - x(t)^2)^2)*(G^2 - x(t)^2)/(2*tan(alpha)*(G__1^2 - x(t)^2)) + N__f*epsilon__0*L*(-2*x(t)/(G^2 - x(t)^2) + 2*(G__2^2 - x(t)^2)*x(t)/(G^2 - x(t)^2)^2)*(G^2 - x(t)^2)/(2*tan(alpha)*(G__2^2 - x(t)^2)))/(N__f*epsilon__0*L*ln((G__1^2 - x(t)^2)/(G^2 - x(t)^2))/(2*tan(alpha)) + N__f*epsilon__0*L*ln((G__2^2 - x(t)^2)/(G^2 - x(t)^2))/(2*tan(alpha)) + C__p) - d*diff(x(t), t), diff(q(t), t) = (q(t)/(N__f*epsilon__0*L*ln((G__1^2 - x(t)^2)/(G^2 - x(t)^2))/(2*tan(alpha)) + N__f*epsilon__0*L*ln((G__2^2 - x(t)^2)/(G^2 - x(t)^2))/(2*tan(alpha)) + C__p) + V__bias)/R1};
 (1)
 > indets(odeA);
 (2)
 > params:={A, C__p, G, G__1, G__2, L, N__f, R1, V__bias, alpha, d, f, k, m,  epsilon__0};
 (3)
 > n:=nops(params); Sol:=dsolve({eval(odeA,params=~{\$ 1..n})[],x(0)=1,q(0)=2,D(x)(0)=0}, numeric);
 (4)
 > plots:-odeplot(Sol,[[t,x(t)],[t,q(t)]], t=0..7,color=[red,blue]);
 >

## explicit...

You do not need to use  plots:-implicitplot  command because one variable is easily expressed through another one  mu=ln(1+x)/x . Immediately we get a smooth curve without any additional options. Here are 2 variants for plotting:

```restart;
plot([ln(1+x)/x, x, x = -1 .. 5], color = black, labels = [mu, x], view = [0 .. 5, -1 .. 5]);  # x as a function of mu
plot(ln(1+x)/x, x = -1 .. 5, color = black, labels = [x, mu], view = [-1 .. 5, 0 .. 5]); # mu as a function of x ```

## Recursive sequence...

I do not understand what the problem is. Maple handles this task easily:

 > restart; F := rsolve({16*s(n+1) = 2+12*s(n)-2*s(n-1), s(1) = 1, s(2) = 5/8}, s); s:=unapply(F, n); seq(s(n), n=1..10); plot([seq([n,s(n)], n=1..10)], style=point, symbol=solidcircle, color=red,size=[1000,300], view=[0..10,0..1]); limit(s(n), n=infinity);
 (1)
 > # Without rsolve restart; s:=proc(n) option remember; if n=1 then return 1 elif n=2 then return 5/8 else 1/8+3*s(n-1)*(1/4)-(1/8)*s(n-2) fi; end proc:
 > seq(s(n), n=1..10); plot([seq([n,s(n)], n=1..10)], style=point, symbol=solidcircle, color=red,size=[1000,300], view=[0..10,0..1]);
 >

Unfortunately, Optimization:-Maximize command in this example returns an erroneous result (I use Maple 2018.2), since it returns only a local extremum. For the correct solution, it is useful to plot graphs and use  initialpoint  option:

 > restart;
 > Optimization:-Maximize(2*x^2 + 2*y^2 + y, {2*x + y <= 6, y^2 - x <= 2});   # This is an incorrect result plots:-inequal({2*x + y <= 6, y^2 - x <= 2},x=-3..5,y=-3..3); solve({2*x + y = 6, y^2 - x = 2}); plot3d(2*x^2 + 2*y^2 + y, y=-5/2..2,x=(6-y)/2..y^2-2, axes=normal); Optimization:-Maximize(2*x^2 + 2*y^2 + y, {2*x + y <= 6, y^2 - x <= 2}, initialpoint=[x=3,y=-2]);  # OK eval(2*x^2 + 2*y^2 + y,[x = 17/4, y = -5/2]); # Symbolic result evalf(%);
 (1)
 >

Edit.

## eval...

As a workaround use the  eval  command:

```restart;
f(x):=a+b;
a:=5;
b:=2;
f(x):=eval(f(x));```

Edit. In addition to the acer's advice, you can also use indexed names:

```restart;
f[x]:=a+b;
a:=5;
b:=2;
f[x]:=f[x];
```

gamma  is a protected constant in Maple, and  S  cannot be differentiated by a constant. Execute it first

local gamma;

## Proof...

We have

f = 2*x^5-x^3*y+2*x^2*y^2-x*y^3+2*y^5 = 2*(x^5+y^5) - x*y*(x^2+y^2-2*x*y) = 2*(x^5+y^5) - x*y*(x-y)^2

If  x<0  and  y<0  then both summands are negative.

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