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dharr

Dr. David Harrington

8697 Reputation

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21 years, 80 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

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Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

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These are answers submitted by dharr

Converting to a first order system...

dharr 8697
2 hours ago
0 1

I didn't undertand what you said. It looks as though you were trying to manually make a first order system. I fixed this and get the same result as convertsys.

systems.mw

bad parameter values...

dharr 8697
Yesterday at 8:58 AM
2 5
> 

restart
> 

eq1 := S__1 = sqrt((-beta[1]+sqrt(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2))/beta[2]); eq2 := S__2 = (1/2)*sqrt(-(2*(beta[1]+sqrt(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)))/beta[2]); eq3 := S__3 = sqrt(2)*sqrt(chi*(4*chi^2*k^2*p+4*chi^2*w+2*chi*p*beta[1]-p^2*beta[2]))/(4*chi^2); eq4 := T__1 = sqrt(-2*chi*p)/(2*chi); eqs := {eq1, eq2, eq3, eq4}

S__1 = ((-beta[1]+(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2))/beta[2])^(1/2)

S__2 = (1/2)*(-2*(beta[1]+(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2))/beta[2])^(1/2)

S__3 = (1/4)*2^(1/2)*(chi*(4*chi^2*k^2*p+4*chi^2*w+2*chi*p*beta[1]-p^2*beta[2]))^(1/2)/chi^2

T__1 = (1/2)*(-2*chi*p)^(1/2)/chi

{S__1 = ((-beta[1]+(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2))/beta[2])^(1/2), S__2 = (1/2)*(-2*(beta[1]+(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2))/beta[2])^(1/2), S__3 = (1/4)*2^(1/2)*(chi*(4*chi^2*k^2*p+4*chi^2*w+2*chi*p*beta[1]-p^2*beta[2]))^(1/2)/chi^2, T__1 = (1/2)*(-2*chi*p)^(1/2)/chi}

> 

indets(eqs)

{S__1, S__2, S__3, T__1, chi, k, p, w, beta[1], beta[2], (chi*(4*chi^2*k^2*p+4*chi^2*w+2*chi*p*beta[1]-p^2*beta[2]))^(1/2), ((-beta[1]+(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2))/beta[2])^(1/2), (-2*chi*p)^(1/2), (-2*(beta[1]+(4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2))/beta[2])^(1/2), (4*k^2*p*beta[2]+4*w*beta[2]+beta[1]^2)^(1/2)}

We have 4 eqns, so choose 4  variables to solve for

> 

sol := solve(eqs, {p, w, beta[1], beta[2]})

{p = -2*T__1^2*chi, w = 2*chi*(S__1^2*S__2^2*T__1^2*k^2-S__1^2*T__1^4*k^2-2*S__2^2*T__1^4*k^2+2*T__1^6*k^2+S__1^2*S__2^2*S__3^2)/(S__1^2*S__2^2-S__1^2*T__1^2-2*S__2^2*T__1^2+2*T__1^4), beta[1] = 2*S__3^2*chi*(S__1^2+2*S__2^2)/(S__1^2*S__2^2-S__1^2*T__1^2-2*S__2^2*T__1^2+2*T__1^4), beta[2] = -4*S__3^2*chi/(S__1^2*S__2^2-S__1^2*T__1^2-2*S__2^2*T__1^2+2*T__1^4)}

Put values in - we see a problem

> 

eval(sol, {S__1 = 1, S__2 = 1, S__3 = 1, T__1 = 1, k = 1})

Error, numeric exception: division by zero

For some other values it should work

> 

eval(sol, {S__1 = 1, S__2 = 1, S__3 = 1, T__1 = 2, k = 1})

{p = -8*chi, w = (170/21)*chi, beta[1] = (2/7)*chi, beta[2] = -(4/21)*chi}

NULL

Download find-parameter.mw

Bug...

dharr 8697
November 12 2025
0 0

Someone else found a similar problem here and I reported this bug to Maple. The same workaround might work here - put simplify(...) around the expression with the square root.

output=Array...

dharr 8697
November 11 2025
1 4

You can use the output=Array option of dsolve to do this.

> 

restart

Specify required x values

> 

xpts := Array([seq(0 .. 3.0, numelems = 20)])

Array(%id = 36893490082029227172)

> 

ode := {diff(y(x), x) = -a*y(x), y(0) = 1}

{diff(y(x), x) = -a*y(x), y(0) = 1}

> 

NULL
> 

ans20 := dsolve(eval(ode, a = 2.0), numeric, output = xpts); ans21 := dsolve(eval(ode, a = 2.1), numeric, output = xpts)

extract vectors of x and y values

> 

xx := ans20[2, 1][() .. (), 1]; y20 := ans20[2, 1][() .. (), 2]; y21 := ans21[2, 1][() .. (), 2]

Plot the difference

> 

plot(xx, y21-y20)

Plot the ratio (enter \/~)

> 

plot(xx, `~`[`/`](y21, y20))

Same thing using the new (Maple 2024) elementwise

> 

plot(xx, elementwise(y21/y20))

> 

NULL

Download dsolveArray.mw

Integration constant...

dharr 8697
November 09 2025
4 3

If you expand Maple's result you get (1/4)*x^4 + x^2 + 1, differing from your expected value by a constant. Since you and Maple are both omitting arbitrary constants, both answers are correct.

Number of BCs...

dharr 8697
November 08 2025
0 1

@KIRAN SAJJAN I would write the 6th bc as (D@@2)(f)(1) = 0. But the equations have a third order derivative of f and a second order derivative of theta, so Maple is right in only expecting 5 boundary conditions. So the paper has an error, perhaps in the odes or perhaps in the bcs (redundant one perhaps?, but I didn't check.)

I recommend putting the parameters in a set so you can look at the odes without the numbers in to check them. Then it is easier to find typos. I found two. The second lambda_val just before Bstar in the second equation should be omitted, and the first equation has an incorrect term. Corrected here:

thin_film_base_paper_comparision.mw

Successive substitutions...

dharr 8697
November 08 2025
2 6

[Updated] If I understand you correctly, then this classic method should work.

> 

restart;
> 

with(StringTools):
> 

s:="from -\\infty to \\infty and then from \\infty to -\\infty";

"from -\infty to \infty and then from \infty to -\infty"

> 

SubstituteAll(s, "-\\infty", "#@#"):
SubstituteAll(%, "\\infty", "+\\infty"):
s2:=SubstituteAll(%, "#@#","-\\infty");

"from -\infty to +\infty and then from +\infty to -\infty"

> 

NULL

Download Substitute.mw

No polynomial solutions...

dharr 8697
November 04 2025
2 0

The Heunc help page has some detailed instructions for finding polynomial solutions.

> 

restart;
> 

local gamma:
> 

Hc := HeunC(-4*b, 4*b, 2, 8*b^2, -8*b^2 - _c[2] + 1, (-r + 2*b)/(2*b));

HeunC(-4*b, 4*b, 2, 8*b^2, -8*b^2-_c[2]+1, (1/2)*(-r+2*b)/b)

> 

alpha, beta, gamma, delta, eta := op(1..5, Hc);

-4*b, 4*b, 2, 8*b^2, -8*b^2-_c[2]+1

A necessary condition for a polynomial solution of degree n is (see the HeunC help page)

> 

poly_cond := delta = -(n + (gamma + beta + 2)/2)*alpha;

8*b^2 = 4*(n+2+2*b)*b

So b has to be (for any n)

> 

solve(poly_cond, b);

0

But the last argument has b in the denominator, so b=0 does not work, and there can be no polynomial solutions

> 

eval(Hc,b=0);

Error, numeric exception: division by zero

> 

NULL

Download polysols.mw

no solution...

dharr 8697
November 02 2025
1 2

There doesn't seem to be a solution, assuming the data you give is exactly correct. The presence of 1+A*3.7653333333333333333*10^(-18)*nu needs very large A for the types of nu values you have, otherwise the A*c*nu term is too small compared to 1 to make a difference.

MLsolve.mw

For the followup examples...

dharr 8697
November 02 2025
1 5

[Edit: I give a more general answer below.]

> 

restart;
> 

local gamma;

gamma

If we know that a is in the field generated by b and we have each of them written as RootOf  a single polynomial

> 

a := RootOf(4*_Z^3 - 12*_Z^2 - 24*_Z - 25, index = 1); # which actually equals “(b + 1)^2”, but assuming we do not know this beforehand
b := RootOf(2*_Z^3 - 3, index = 1);

RootOf(4*_Z^3-12*_Z^2-24*_Z-25, index = 1)

RootOf(2*_Z^3-3, index = 1)

> 

alias(alpha=a,beta=b);

alpha, beta

Then we "just" evaluate a specifying b as a field extension. Unfortunately, many of the evala subcommands such as Normal deduce the field extension from the coefficients, and don't allow us to specify it. Factor allows us to specify it, and even though we don't need to actually factor here it will serve the purpose:

> 

evala(Factor(a,b));

beta^2+2*beta+1

We don't need Primfield, which just confirms that a can be written in terms of b, so the field extension can just be b

> 

evala(Primfield({a,b}));

[[beta = beta], [beta = beta]]

Algfield works similarly, but gives the relationship between a and b

> 

evala(Algfield({a,b}));

[[alpha = beta^2+2*beta+1], [], {beta}, true, {}]

In the second case, Maple already simplifies each of these in terms of RootOf(_Z^2-2, index = 1) and RootOf(_Z^2+1, index = 1), which doesn't help us here.

> 

c:= convert((-1)^(1/4), RootOf);
evalf(%);
d:= convert((sqrt(2) - 1)*I, RootOf); # which actually equals c*(c^2 - c + 1), but assuming we do not know this beforehand
evalf(%);

(1/2)*RootOf(_Z^2-2, index = 1)+(1/2)*RootOf(_Z^2-2, index = 1)*RootOf(_Z^2+1, index = 1)

.7071067810+.7071067810*I

RootOf(_Z^2+1, index = 1)*(RootOf(_Z^2-2, index = 1)-1)

.4142135620*I

Convert these to forms that are RootOf a single polynomial

> 

mp:=evala(Minpoly(c,x)):
c:=select(q->evala(c-q)=0,[seq(RootOf(mp,index=i),i=1..degree(mp,x))])[];
evalf(%);
mp:=evala(Minpoly(d,x)):
d:=select(q->evala(d-q)=0,[seq(RootOf(mp,index=i),i=1..degree(mp,x))])[];
evalf(%);

RootOf(_Z^4+1, index = 1)

.7071067812+.7071067812*I

RootOf(_Z^4+6*_Z^2+1, index = 1)

.4142135624*I

> 

alias(gamma=c, delta=d);

alpha, beta, gamma, delta

> 

evala(Factor(d,c));

gamma^3-gamma^2+gamma

In this example, we can write RootOf(_Z^4+6*_Z^2+1) in terms of  RootOf(_Z^4+1, index = 1) The method doesn't work the other way around - it just returns c:

> 

evala(Factor(c,d));

gamma

Same problem for the first one if we want to do it the other way around.

So this method also does not work on the example in the original question

> 

evala(Factor(b,a));

beta

> 

NULL

Download evala.mw

Child...

dharr 8697
October 25 2025
3 1

Perhaps I missed something since age 21 in 1968 seems beyond a child. Not a very elegant solution.

"During a birthday party, the birthday child realizes: In 1968, I was the same age as the sum of the digits of my birth year. How old will I be now at the end of 2025?"

> 

restart

Birth year and its sum of digits

> 

b := 1900+10*d1+d2; s := 1+9+d1+d2

1900+10*d1+d2

10+d1+d2

Condition

> 

eq := 1968-b = s

68-10*d1-d2 = 10+d1+d2

Just search

> 

for d1 from 0 to 9 do for d2 from 0 to 9 do if eq then print(d1, d2, eval(2025-b)) end if end do end do

4, 7, 78

So born in 1947, age 21 in 1968 = 1+9+4+7; age in 2025:

> 

2025-1947

78

> 

NULL

Download Birthday.mw

workaround...

dharr 8697
October 23 2025
1 1

It works using the simplify command rather than the context menu:

restartNULL

 

  k := 10.*Unit('N'/'m')

m := 2.*Unit('kg')NULL

omega := sqrt(k/m)"(->)"2.236067977*Units:-Unit(1/s)

 

"x(t):=A*cos(omega*t)+B*sin(omega*t):"

"v(t):=-A*omega*sin(omega*t)+B*omega*cos(omega*t):"

NULL

L1 := x(0*Unit('s')) = .15*Unit('m')

L2 := v(0*Unit('s')) = 0*Unit('m'/'s')

NULL

s := fsolve({L1, L2}, {A, B}) = {A = .15*Units:-Unit(m), B = -0.}NULL

assign(s)

 

x(t)"(=)".15*Units:-Unit(m)*cos(2.236067977*t*Units:-Unit(1/s))

v(t)

-.3354101966*Units:-Unit(m)*Units:-Unit(1/s^2)^(1/2)*sin(2.236067977*Units:-Unit(1/s^2)^(1/2)*t)

simplify(v(t))

-.3354101966*Units:-Unit(m/s)*sin(2.236067977*t*Units:-Unit(1/s))

v(t)"(=)"-.3354101966*Units:-Unit(m)*sin(2.236067977*t*Units:-Unit(1/s))

``

Download Error_with_units.mw

same with O...

dharr 8697
October 21 2025
1 3

The change to upright font for D and O appears to be hardcoded into the last step of preparing the output, so diificult to fix (at least for users). 

> 

restart
> 

_local(D, O)

O

> 

a*D*O/sin(D)

a*D*O/sin(D)

The typesetting code for this.
Notice that both a and D are inside Typesetting:-mi(), which is for italics

> 

Typesetting:-Typeset(a*D*O/sin(D)); lprint(%)

a*D*O/sin(D)

Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi("a"),Typesetting:-mo(
"⁢"),Typesetting:-mi("D"),Typesetting:-mo("⁢"),
Typesetting:-mi("O")),Typesetting:-mrow(Typesetting:-mi("sin",fontstyle =
"normal"),Typesetting:-mo("⁡"),Typesetting:-mfenced(Typesetting:-
mi("D"))))

Note different output here, suggesting that only in the final stage is there code to force D and O upright.

> 

Typesetting:-mi("D"); Typesetting:-mi("O"); Typesetting:-mi(a)

D

O

a

> 

NULL

Download Typesetting.mw

assuming real...

dharr 8697
October 21 2025
1 0

adding "assuming real" is a workaround for this case.

> 

interface(version);

`Standard Worksheet Interface, Maple 2025.1, Windows 11, June 12 2025 Build ID 1932578`

> 

Physics:-Version();

`The "Physics Updates" package is not available for the version of Maple under development`

> 

SupportTools:-Version();

`The Customer Support Updates package is not installed. To install it, please run the command SupportTools:-Update().`

> 

 

Example fail when using _EnvAllSolutions := true:

 
> 

restart;

> 

kernelopts('assertlevel'=2):

> 

ode:=diff(y(x),x)=ln(1+y(x)^2);
IC:=y(0)=0;
x0:=0;
sol:=y(x) = -sqrt(-1 + exp(RootOf(-Intat(-1/(2*tau*sqrt(-1 + exp(tau))*exp(-tau)), tau = _Z) + x + _C2)));

eq:=0=eval(rhs(sol),x=x0);
_EnvAllSolutions := true:
_EnvExplicit := true:
sol_C:=_C2=solve(eq,_C2);
sol:=eval(sol,sol_C);
odetest(sol,[ode,IC]) assuming real;
simplify(%, symbolic);

diff(y(x), x) = ln(1+y(x)^2)

y(0) = 0

0

y(x) = -(-1+exp(RootOf(-Intat(-(1/2)/(tau*(-1+exp(tau))^(1/2)*exp(-tau)), tau = _Z)+x+_C2)))^(1/2)

0 = -(-1+exp(RootOf(-Intat(-(1/2)/(tau*(-1+exp(tau))^(1/2)*exp(-tau)), tau = _Z)+_C2)))^(1/2)

_C2 = Intat(-(1/2)*exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = (2*I)*Pi*_Z1)

y(x) = -(-1+exp(RootOf(-Intat(-(1/2)/(tau*(-1+exp(tau))^(1/2)*exp(-tau)), tau = _Z)+x+Intat(-(1/2)*exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = (2*I)*Pi*_Z1))))^(1/2)

[RootOf(Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = _Z)+2*x-Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = (2*I)*Pi*_Z1))-ln(exp(RootOf(Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = _Z)+2*x-Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = (2*I)*Pi*_Z1)))), 0]

[0, 0]

> 

 

Example Works  when using _EnvAllSolutions := false:

 
> 

restart;

> 

kernelopts('assertlevel'=2):

> 

ode:=diff(y(x),x) = ln(1+y(x)^2);
IC:=y(0)=0;
x0:=0;
sol:=y(x) = -sqrt(-1 + exp(RootOf(-Intat(-1/(2*tau*sqrt(-1 + exp(tau))*exp(-tau)), tau = _Z) + x + _C2)));

eq:=0=eval(rhs(sol),x=x0);
_EnvAllSolutions := false:
_EnvExplicit := true:
sol_C:=_C2=solve(eq,_C2);
sol:=eval(sol,sol_C);
odetest(%,[ode,IC])

diff(y(x), x) = ln(1+y(x)^2)

y(0) = 0

0

y(x) = -(-1+exp(RootOf(-Intat(-(1/2)/(tau*(-1+exp(tau))^(1/2)*exp(-tau)), tau = _Z)+x+_C2)))^(1/2)

0 = -(-1+exp(RootOf(-Intat(-(1/2)/(tau*(-1+exp(tau))^(1/2)*exp(-tau)), tau = _Z)+_C2)))^(1/2)

_C2 = Intat(-(1/2)*exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = 0)

y(x) = -(-1+exp(RootOf(-Intat(-(1/2)/(tau*(-1+exp(tau))^(1/2)*exp(-tau)), tau = _Z)+x+Intat(-(1/2)*exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = 0))))^(1/2)

[RootOf(Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = _Z)+2*x-Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = 0))-ln(exp(RootOf(Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = _Z)+2*x-Intat(exp(tau)/(tau*(-1+exp(tau))^(1/2)), tau = 0)))), 0]
> 

 

> 

 

Download odetest_fail_when_using_envAllsol_maple_2025_1_oct_21_2025.mw

collect after substitution...

dharr 8697
October 21 2025
2 5

Is this what you want? Note that when using collect, collecting wrt G{xi) handles negative powers so you do not need to collect wrt 1/G(xi).

> 

restart
> 

with(PDEtools)
> 

undeclare(prime, quiet)
> 

declare(U(xi), quiet)
> 

NULL
> 

ode := -2*(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*c-(108*c*(1/49)+8*U(xi)^2+1)*(diff(diff(U(xi), xi), xi))+18*U(xi)*(diff(U(xi), xi))^2+2*U(xi)*(-17*U(xi)^2*(1/49)+243*c*(1/2401)+9/98+w)

-2*(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*c-((108/49)*c+8*U(xi)^2+1)*(diff(diff(U(xi), xi), xi))+18*U(xi)*(diff(U(xi), xi))^2+2*U(xi)*(-(17/49)*U(xi)^2+(243/2401)*c+9/98+w)

> 

S := alpha[0]+alpha[1]*(diff(G(xi), xi))/G(xi)+beta[0]/G(xi)

alpha[0]+alpha[1]*(diff(G(xi), xi))/G(xi)+beta[0]/G(xi)

> 

WW := diff(G(xi), `$`(xi, 2)) = -lambda*G(xi)+mu

diff(diff(G(xi), xi), xi) = -lambda*G(xi)+mu

> 

NULL
> 

D1 := diff(S, xi)
> 

D11 := subs(WW, D1)
> 

D2 := diff(D11, xi)
> 

D22 := subs(WW, D2)
> 

D3 := diff(D22, xi)
> 

D33 := subs(WW, D3)
> 

D4 := diff(D33, xi)
> 

D44 := subs(WW, D4)
> 

K := U(xi) = S
> 

K1 := diff(U(xi), xi) = D11
> 

K2 := diff(diff(U(xi), xi), xi) = D22
> 

K3 := diff(diff(U(xi), xi), xi, xi) = D33
> 

K4 := diff(diff(U(xi), xi), xi, xi, xi) = D44
> 

P := eval(ode, {K, K1, K2, K3, K4})

-2*(alpha[1]*lambda^2*(diff(G(xi), xi))/G(xi)+15*alpha[1]*lambda*(-lambda*G(xi)+mu)*(diff(G(xi), xi))/G(xi)^2-20*alpha[1]*lambda*(diff(G(xi), xi))^3/G(xi)^3-60*alpha[1]*(-lambda*G(xi)+mu)*(diff(G(xi), xi))^3/G(xi)^4+30*alpha[1]*(-lambda*G(xi)+mu)^2*(diff(G(xi), xi))/G(xi)^3+24*alpha[1]*(diff(G(xi), xi))^5/G(xi)^5+24*beta[0]*(diff(G(xi), xi))^4/G(xi)^5-36*beta[0]*(-lambda*G(xi)+mu)*(diff(G(xi), xi))^2/G(xi)^4-8*beta[0]*lambda*(diff(G(xi), xi))^2/G(xi)^3+6*beta[0]*(-lambda*G(xi)+mu)^2/G(xi)^3+beta[0]*lambda*(-lambda*G(xi)+mu)/G(xi)^2)*c-((108/49)*c+8*(alpha[0]+alpha[1]*(diff(G(xi), xi))/G(xi)+beta[0]/G(xi))^2+1)*(-alpha[1]*lambda*(diff(G(xi), xi))/G(xi)-3*alpha[1]*(-lambda*G(xi)+mu)*(diff(G(xi), xi))/G(xi)^2+2*alpha[1]*(diff(G(xi), xi))^3/G(xi)^3+2*beta[0]*(diff(G(xi), xi))^2/G(xi)^3-beta[0]*(-lambda*G(xi)+mu)/G(xi)^2)+18*(alpha[0]+alpha[1]*(diff(G(xi), xi))/G(xi)+beta[0]/G(xi))*(alpha[1]*(-lambda*G(xi)+mu)/G(xi)-alpha[1]*(diff(G(xi), xi))^2/G(xi)^2-beta[0]*(diff(G(xi), xi))/G(xi)^2)^2+2*(alpha[0]+alpha[1]*(diff(G(xi), xi))/G(xi)+beta[0]/G(xi))*(-(17/49)*(alpha[0]+alpha[1]*(diff(G(xi), xi))/G(xi)+beta[0]/G(xi))^2+(243/2401)*c+9/98+w)

We want to find coefficients G^(n)/G^m. collect(P, {1/G(xi), diff(G(xi), xi)}, distributed)collects the way we want, but coeffs does not pick them off correctly (probably because it wasn't intended to work with diff.)

Write Gp for the derivative, and then collect (distributed) on the result.

> 

P2 := subs(diff(G(xi), xi) = Gp, P)
> 

P3 := collect(P2, {Gp, G(xi)}, distributed)
> 

eqs := {coeffs(P3, {Gp, G(xi)}, 'monomials')}

{2*beta[0]^3, -14*alpha[0]*alpha[1]^2, 6*beta[0]^2*alpha[1], 8*beta[0]^3*mu, 4*beta[0]^2*alpha[1]*mu, 4*alpha[0]*alpha[1]^2*lambda-(102/49)*alpha[0]*alpha[1]^2, 18*alpha[0]*alpha[1]^2*lambda^2+2*alpha[0]*(-(17/49)*alpha[0]^2+(243/2401)*c+9/98+w), -80*alpha[1]*lambda*c-2*((108/49)*c+8*alpha[0]^2+1)*alpha[1]+20*alpha[1]^3*lambda-(34/49)*alpha[1]^3, -32*alpha[1]*lambda^2*c-2*((108/49)*c+8*alpha[0]^2+1)*alpha[1]*lambda+18*alpha[1]^3*lambda^2-(68/49)*alpha[0]^2*alpha[1]+2*alpha[1]*(-(17/49)*alpha[0]^2+(243/2401)*c+9/98+w), -60*alpha[1]*mu^2*c+4*beta[0]^2*alpha[1]*lambda+28*beta[0]*alpha[0]*alpha[1]*mu+18*alpha[1]^3*mu^2-(102/49)*beta[0]^2*alpha[1], -56*beta[0]*lambda*c-2*((108/49)*c+8*alpha[0]^2+1)*beta[0]+32*beta[0]*alpha[1]^2*lambda+12*alpha[0]*alpha[1]^2*mu-(102/49)*beta[0]*alpha[1]^2, -12*beta[0]*mu^2*c-8*beta[0]^3*lambda+16*beta[0]^2*alpha[0]*mu+18*beta[0]*alpha[1]^2*mu^2-(34/49)*beta[0]^3, 90*alpha[1]*lambda*mu*c-12*beta[0]*alpha[0]*alpha[1]*lambda+3*((108/49)*c+8*alpha[0]^2+1)*alpha[1]*mu-36*alpha[1]^3*mu*lambda-(204/49)*beta[0]*alpha[0]*alpha[1], -10*beta[0]*lambda^2*c-((108/49)*c+8*alpha[0]^2+1)*beta[0]*lambda+18*beta[0]*alpha[1]^2*lambda^2-36*alpha[0]*alpha[1]^2*mu*lambda+2*beta[0]*(-(17/49)*alpha[0]^2+(243/2401)*c+9/98+w)-(68/49)*alpha[0]^2*beta[0], 22*beta[0]*mu*lambda*c-16*beta[0]^2*alpha[0]*lambda+((108/49)*c+8*alpha[0]^2+1)*beta[0]*mu-36*beta[0]*alpha[1]^2*mu*lambda+18*alpha[0]*alpha[1]^2*mu^2-(102/49)*beta[0]^2*alpha[0], 2*alpha[1]^3-48*c*alpha[1], 6*alpha[1]^2*beta[0]-48*c*beta[0], -16*mu*alpha[1]^2*beta[0]+72*c*mu*beta[0]-14*alpha[0]*beta[0]^2, -12*mu*alpha[1]^3+120*c*mu*alpha[1]-28*alpha[0]*alpha[1]*beta[0]}

> 

monomials

1, Gp^4/G(xi)^4, Gp^4/G(xi)^5, Gp^3/G(xi)^3, Gp^3/G(xi)^4, Gp^3/G(xi)^5, Gp^2/G(xi)^2, Gp^2/G(xi)^3, Gp^2/G(xi)^4, Gp^2/G(xi)^5, Gp/G(xi), Gp/G(xi)^2, Gp/G(xi)^3, Gp/G(xi)^4, 1/G(xi), 1/G(xi)^2, 1/G(xi)^3, 1/G(xi)^4, Gp^5/G(xi)^5

> 

``

NULL

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