## remove RootOf, after solving...

Hy

How I solved it accurately,and remove "'rootoff

eq1 := alpha + beta*r[c] - d*n[c] - Upsilon*n[c]*(n[r] + r[c]) - n[r]*(alpha - d*n[c] - b*(n[r] + r[c]));
q2 := e&Upsi;*n[c]*(n[r] + r[c]) - mu*n[r] + d*n[c]*n[r] + b*n[c]*n[r] - alpha*n[r];
eq3 := b*n[c]*n[r] + d*n[c]*n[r] - alpha*n[r] - beta*r[c] + mu*n[r];
eq1 := alpha + beta r[c] - d n[c] - Upsilon n[c] (n[r] + r[c])

- n[r] (alpha - d n[c] - b (n[r] + r[c]))
eq2 := Upsilon n[c] (n[r] + r[c]) - mu n[r] + d n[c] n[r]

+ b n[c] n[r] - alpha n[r]
eq3 := b n[c] n[r] + d n[c] n[r] - alpha n[r] - beta r[c]

+ mu n[r]
solve({eq1, eq2, eq3}, {n[c], n[r], r[c]});
/       alpha                    \    /             /
{ n[c] = -----, n[r] = 0, r[c] = 0 }, { n[c] = RootOf\(Upsilon b
\         d                      /    \

2
+ Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu

\               //
+ b beta + beta d) _Z - alpha beta - mu beta/, n[r] = RootOf\\

2           2                         \   2   /
b  beta + 2 b  mu + b beta d + 2 b d mu/ _Z  + \
/                          2
-RootOf\(Upsilon b + Upsilon d) _Z  + (-Upsilon alpha

+ Upsilon beta + Upsilon mu + b beta + beta d) _Z - alpha beta

\                           /
- mu beta/ Upsilon alpha b - 2 RootOf\(Upsilon b + Upsilon d)

2
_Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu + b beta

\                          /
+ beta d) _Z - alpha beta - mu beta/ Upsilon b beta - 2 RootOf\

2
(Upsilon b + Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta

\
+ Upsilon mu + b beta + beta d) _Z - alpha beta - mu beta/

/                          2
Upsilon b mu - 3 RootOf\(Upsilon b + Upsilon d) _Z  + (
-Upsilon alpha + Upsilon beta + Upsilon mu + b beta + beta d) _Z

\                          /
- alpha beta - mu beta/ Upsilon beta d - 3 RootOf\(Upsilon b

2
+ Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu

\
+ b beta + beta d) _Z - alpha beta - mu beta/ Upsilon d mu

\            /
- alpha b beta + 2 b beta mu + 3 beta d mu/ _Z + RootOf\

2
(Upsilon b + Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta

\
+ Upsilon mu + b beta + beta d) _Z - alpha beta - mu beta/

/                          2
Upsilon alpha b + RootOf\(Upsilon b + Upsilon d) _Z  + (
-Upsilon alpha + Upsilon beta + Upsilon mu + b beta + beta d) _Z

\                        /
- alpha beta - mu beta/ Upsilon beta d + RootOf\(Upsilon b

2
+ Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu

\
+ b beta + beta d) _Z - alpha beta - mu beta/ Upsilon d mu

\          1   /      // 2
+ alpha b beta - beta d mu/, r[c] = ---- \RootOf\\b  beta
beta

2                         \   2   /
+ 2 b  mu + b beta d + 2 b d mu/ _Z  + \
/                          2
-RootOf\(Upsilon b + Upsilon d) _Z  + (-Upsilon alpha

+ Upsilon beta + Upsilon mu + b beta + beta d) _Z - alpha beta

\                           /
- mu beta/ Upsilon alpha b - 2 RootOf\(Upsilon b + Upsilon d)

2
_Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu + b beta

\                          /
+ beta d) _Z - alpha beta - mu beta/ Upsilon b beta - 2 RootOf\

2
(Upsilon b + Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta

\
+ Upsilon mu + b beta + beta d) _Z - alpha beta - mu beta/

/                          2
Upsilon b mu - 3 RootOf\(Upsilon b + Upsilon d) _Z  + (
-Upsilon alpha + Upsilon beta + Upsilon mu + b beta + beta d) _Z

\                          /
- alpha beta - mu beta/ Upsilon beta d - 3 RootOf\(Upsilon b

2
+ Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu

\
+ b beta + beta d) _Z - alpha beta - mu beta/ Upsilon d mu

\            /
- alpha b beta + 2 b beta mu + 3 beta d mu/ _Z + RootOf\

2
(Upsilon b + Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta

\
+ Upsilon mu + b beta + beta d) _Z - alpha beta - mu beta/

/                          2
Upsilon alpha b + RootOf\(Upsilon b + Upsilon d) _Z  + (
-Upsilon alpha + Upsilon beta + Upsilon mu + b beta + beta d) _Z

\                        /
- alpha beta - mu beta/ Upsilon beta d + RootOf\(Upsilon b

2
+ Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu

\
+ b beta + beta d) _Z - alpha beta - mu beta/ Upsilon d mu

\ /      /
+ alpha b beta - beta d mu/ \RootOf\(Upsilon b + Upsilon d)

2
_Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu + b beta

\           /
+ beta d) _Z - alpha beta - mu beta/ b + RootOf\(Upsilon b

2
+ Upsilon d) _Z  + (-Upsilon alpha + Upsilon beta + Upsilon mu

\               \\
+ b beta + beta d) _Z - alpha beta - mu beta/ d - alpha + mu//

\
}
/

## critical point of f(x,y)...

Dear all

I have  a funciton f(x,y)
I compute the critical point, maple gives me three different points, denoted in the code tmin

from these points how can I fix one  one of them that  give me the min( f(x))

code.mw

thank you for any help

## how to compute symbolic Expressions ...

H all expert

first this Expression

ni := diff(Q(x, t), t)+a*Q*(x, t)*(diff(Q(x, t), x))+b*(diff(Q(x, t), t$3))+d*(diff(Q(x, t), t$5)) = 0

then I want to solve  diff(Q(x, t), t$5) diff(Q(x, t), t$5) = solve(ni, diff(Q(x, t), t$5))#eq2 p (x, t) :=H1(t)*Q(x*H2(t), H3(t)) #assumption k := diff(p(x, t), t)+a*p(x, t)*(diff(p(x, t), x))+diff(p(x, t), t$3)+d*(diff(p(x, t), t$5))+c*p(x, t) #eq3 r := subs(diff(Q(x, t), t$5), k) #subs eq2 in eq3

i recived error

SUBS1.mw

## subs and algsubs fail to work on noncommutative va...

I need some calculations in a noncommutative ring - specifically, I need to rewrite/simplify some algebraic expressions in those variables.  My variables need to be indexed by integers - for example, I need variables A[n], B[n], and C[n], where n is an arbitrary integer.  These variables should not commute.  My approach to doing this so far has been to use quantumoperators:

with(Physics);
Setup(quantumoperators = {A, B, C});
[quantumoperators = {A, B, C}]

There are some commutator relations that need to be imposed.  For example, suppose we have

Commutator(A[m], B[n]) = C[m+n].

I can impose these commutator relations using

Setup(algebrarules = {%Commutator(A[m], B[n]) = C[m+n]});

That part is ok.   But I would also like to subsitute certain algebraic expressions in these variables.  For example, suppose I would like to substitute A[2]*B[3] = 5. It seems I should be able to do so using subs or algsubs. This works on very simple expressions: for example, I get

algsubs(A[1]*B[2] = 5, A[1]*B[2] + C[3]);
5 + C3

as I would expect.  But the following does not work:

algsubs(A[1]*B[2] = 5, A[1]*B[2]*C[4] + C[3]);
A1B2C4 + C3

But I should get output 5C4+ C3.  This problem also persists if I use subs in stead of  algsubs.  It does not appear if I use 'standard' commuting variables.

## rk-2 for system of odes...

I want to apply rk-2 for the following system of odes. I want to know what is process of rk-2 method maple used? Is it possible that we can see the complete process maple used?

## What are the real-solve commands in Maple?...

Very often it happens that using solve alone, gives huge expressions that can't be used. The simplest thing to do is to wrap it inside an evalf. But then sometimes, even your system only has real solutions, you may get some complex numbers. When this evalf(solve()) being used inside an algorithm, then disastrous consequences may arise! If the system consists of a single equation of a single variable, then you may have some more tools. But if you have a system of several equations in several variables, you have less options. I am mostly interested in polynomials, I know several approaches to use and code to solve and get only the real solutions, but my codes might be not very optimized. In Maple 2022, one predefined command which is nice is RootFinding:-Isolate but it has one issue and it is that this command only likes numeric coefficients which means integers, fraction of integers and float numbers, so no square root or other real numbers of this shape in the coefficients. I thought it might be a good idea to have a list of all solving commands in Maple that only return the real solutions or have the options to restrict to only real solutions. fsolve is not very ideal, because it only returns one solution.

## Plot with explore...

How to plot this equation with explore or animation

E[1]:=Sum((GAMMA(((beta+1)n-gamma(nu-1))+k))/(GAMMA(((beta+1)n-gamma(nu-1)))*GAMMA(rho*k + (nu(1-mu)+mu(2 n+1))))*((omega*t^(p))^(k))/(k!),k=0..5);  E[2] :=Sum((GAMMA((gamma(n+1)-beta *n)+k))/(GAMMA((gamma(n+1)-beta *n))*GAMMA(rho*k + (mu(2 n+1))))*((omega*t^(p))^(k))/(k!),k=0..5);    y(p):=Sum(c*gamma^(n)*t^(nu*(1-mu)+mu+2*mu*n-1)*E[1]+gamma^(n)t^(mu(2 n+1)-1)*E[2]*g, n=0..5);

with the conditions

mu, nu \in (0, 1); omega \in R; rho > 0; gamma, beta > or = 0; c & g are constant

## Short Cut for displaying in 2D input...

How to create a matrix in 2D input without using palette? Any short cut that returns a matrix does not display as a matrix in 2D input.

## Re: How to pick up 'LPSolve' result...

Hello all,

Is there any chance to ask these two questions?

1) What would be the way to pick up the result from 'LPSolve' command? I thought that the command would overlay the result to the input matrix, but I was wrong.

2) This question is not related to Maple, but I saw that the transportation problem described in the attached worksheet has multiple solutions. In other words, the solution that came up with 'LPSolve' command was different from what the textbook presented, but both of them were legitimate solutions. Therefore, I'm wondering if there is a way to test the uniqueness of the solution.

In Kwon Park

Q20220416.mw

## How to get &D; into the favorites palette?...

I got a solution to use D as a symbol that prints well in italic. For convienience I like to have it in my favorite palette.
When I drag &D; from a Maple Input line to the favorites palette, the ampersand and the statement operator are removed. Copy and paste have the same effect on the pasted selection. If  these characters are removed, &D; becomes the differential operator D. That's not what I want.
Is it possible at all to get &D; or simliar expressions using special characters within left single quotes into the favorite palette?

## converting to spherical Bessel...

Hi,

I want to convert BesselJ(n,x) to the spherical version of it.

Is there a defined command in Maple that does this??

For example:

sqrt(Pi)*sqrt(2)*BesselJ(1/2, x)/(2*sqrt(x))

is equivallent to

sin(x)/x

Indeed, the latter is the spherical version of the former one.

However, I can define a function that does it, see:

SBesselJ := proc (n,x) options operator, arrow, function_assign; expand(simplify(sqrt(Pi/(2*x))*BesselJ(n+1/2, x),'assume = positive')) end proc;

but is there a command that converts some terms to our defined function??

## "Dual" command in Logic package...

Dear all,

I do not understand the output of the "Dual" command in the Logic package.

According to the help:

"The Dual command returns the dual of the Boolean expression b, that is, the expression generated by replacing &and with &or, &or with &and, leaving &not fixed, and extending to the remaining Boolean operators by their formulas in terms of &and, &or, and &not."

Applying Dual to "a &implies b" gives:
Logic:-&implies(&not(a), &not(b))

But "a &implies b" is equivalent to "&not(a) &or b" and its Dual would be "&not(a) &and b", which is not equivalent to the result Maple returns.

Best regards,
Anthei

## module()...end module...

Ha := 5; Gr := 2; Pr := 8; b := 4; lambda := 3;
PDE := {diff(theta(x, t), t)-(diff(theta(x, t), x, x))/Pr = 0, diff(u(x, t), t)-(diff(u(x, t), x, x))+Ha*u(x, t)-Gr*theta(x, t) = 0};
IBC := {theta(0, t) = 1+b(1-cos(lambda*t)), theta(1, t) = 1, theta(x, 0) = 0, u(0, t) = 1-cos(t), u(1, t) = 1, u(x, 0) = 0};
{theta(0, t) = 5, theta(1, t) = 1, theta(x, 0) = 0,

u(0, t) = 1 - cos(t), u(1, t) = 1, u(x, 0) = 0}
pds := pdsolve(PDE, IBC, numeric);
module()...end module

how we can expand module()...end module and see solutions of the problem.

## System of ode analytic solution...

How to solve the following system of ode analytically in maple?

diff(s(t), t) = gamma*s(t) + eta*s(t) + sigma*m(t)*s(t) - kappa*s(t) - phi*s(t), diff(g(t), t) = -alpha*g(t)*m(t) + mu*g(t) + delta*s(t)*g(t), diff(m(t), t) = beta*s(t) - alpha_1*g(t)*m(t)