## 13613 Reputation

19 years, 226 days

## An alternative approach...

An approach that doesn't begin by solving for the derivative, but differentiates the ode and thereby turns it into a second order ode:

```restart;

Digits:=15:
V:=n^2*((T[e]+T[i])/T[e])*((((n-1)*(1+1/(2*delta[p]))))-ln(n)-ln(n/(2*delta[p])));
W:=eval(V,{T[e]=6/5,T[i]=1/100,delta[p]=1/5,n=n(x)});
ode:=diff(n(x),x)^2+2*W=0;
############################
eval(convert(ode,D),{n(x)=1});
ICS:=solve(eval(%,x=0),{D(n)(0)});
ic:=n(0)=1;
ODE:=diff(ode,x);
RES1:=dsolve({ODE,op(ICS[1]),ic},numeric);
plots:-odeplot(RES1,[x,n(x)],-10..10);
RES2:=dsolve({ODE,op(ICS[2]),n(0)=1},numeric);
plots:-odeplot(RES2,[x,n(x)],-10..10);
#############################
```

The first plot:

## No worksheet...

You forgot to attach the worksheet showing the result from Maple 11.

## Time...

@acer Thank you! Measuring time[real] I got 16 minutes on my computer.

## How long does it take?...

@acer For how long does RootFinding:-Isolate have to run to produce the results in your worksheet?
I'm using
`Maple 2022.0, X86 64 WINDOWS, Mar 8 2022, Build ID 1599809`

on a reasonably fast and rather new computer.

## Maple 12...

@acer I executed your worksheet in Maple 12 and I got this dreaded result from NLPSolve:

sol := NLPSolve(bandpeak(Zstub), Zstub=Z[1]..Z[N],
method=sqp, initialpoint = [Zstub=Z[2]]);
Warning, no iterations performed as initial point satisfies first-order conditions
[0.375277160248564, [Zstub = 5.02596984406875]]

whereas in Maple 2022 I get

sol := NLPSolve(bandpeak(Zstub), Zstub=Z[1]..Z[N],
method=sqp, initialpoint = [Zstub=Z[2]]);
Warning, limiting number of evaluations reached
...................................many of these Warnings
Warning, limiting number of evaluations reached
[0.111740781972696004, [Zstub = HFloat(43.9058440724061)]]

## Cannot reproduce...

I tried your worksheet and didn't get any errors.
`Maple 2022.0, X86 64 WINDOWS, Mar 8 2022, Build ID 1599809`

## Plural or singular...

@Carl Love I'm always being put off (to use a polite expression) by the use of plural pronouns for a situation, where it is quite obvious a singular version should have been used.
" The member @nm has a long-standing habit of deleting their own Questions"
I would simply have said: The member @nm has a long-standing habit of deleting his own Questions.
By this I'm not implying that nm is a male (I don't know), but just using the time honored usage in such cases.

## Example...

@Mac Dude Could you give us an example?

I have an old one of my own:

```restart;
Digits:=15:
sys := {6*exp(x+y)/(1+exp(x+y))+exp(2*y)/(1+exp(2*y)) = 2,
5*exp(2*x)/(1+exp(2*x))+2*exp(x+y)/(1+exp(x+y)) = 1}:
plots:-implicitplot(sys,x=-10..10,y=-10..10);
S:=solve(sys);
evalf(S[1]);
fsolve(sys,S[1]); # returns unevaluated
fsolve(sys,{x=-1.2..-1, y=-.1..0.1}); #OK
```

But that one is not new, even Maple 12 has that behavior.

## The help page...

@opus64 In the help page for isolate you also find this statement:

"For direct solutions of equations, solve may be more efficient than isolate, which is intended, primarily, for use in an interactive Maple session." (my emphasis).
The code for isolate can easily be seen:

```restart;
showstat(isolate,1..6); # we need only look at the lines 1 through 6.
eq:=P=A+(z+x1/2/z); # Using x1 instead of x
## In the code expr and x will be:
expr:=eq; x:=z+x1/2/z;
has(expr,x); # Line 3, answer false
type(x,`^`);   # Line 4, answer false
```

'has' is builtin, so you cannot see its code:
op(3,eval(has)); # option builtin
but the help page for has states:

"The has(f, x) function returns true if f contains the expression x;  otherwise false is returned.
The expression f contains x if and only if a subexpression of f (as defined by Maple's op function) is equal to x." (my emphasis)

## The poster himself?...

Could it be the poster himself?

## Greek...

I cannot check Maple V Release 5 right now since I don't have it on this machine.
I checked Maple 12, however, and that behaves like all the versions I remember. I have had all Maple releases since Maple V Release 2.
I would really be surprised if Maple V Release 5 (or 5.1) really displays alpha and beta as they are written in Greek.
Could you show us a screenshot from Maple V Release 5?

## The data...

The output from sparsematrixplot makes suggests that what you ask for isn't readily available:

```restart;
with(plots):
with(LinearAlgebra):

A := Matrix([[2, 1, 0, 0, 3], [0, 2, 1, 0, 0], [0, 0, 2, 1, 0], [0, 0, 0, 2, 1], [0, 0, 0, 0, 2]]);

sparsematrixplot(A, matrixview, color = "DarkGreen"); p:=%:
pd:=plottools:-getdata(p);
pd[1];
plot(pd[1,3],thickness=4);
plot(pd[1,3],style=point,symbol=solidcircle,symbolsize=50);
```

## Change of variables...

@vv Maybe I misunderstand you when you say " Unfortunately, Maple cannot compute it."
In Maple 2021.2 I get (using your suggested change of variables):

```A:=Int(2/(1-x^p)^(1/p), x=0..1);
IntegrationTools:-Change(A,x^p=t) assuming p>1;
value(%) assuming p>1;
```

The answer given is 2*Beta(1/p, -1/p + 1)/p, i.e. the same as the result by cook.

So what is the correct answer?

## Make everything concrete...

@escorpsy I don't see any way of getting a symbolic answer with parameters. But you can certainly find exact answers with concrete values for the parameters. As an example:

```M := Int((x*(1 - x)*(-I*epsilon + p^2) + m^2)^((d - 4)/2), x = 0 .. 1);
value(eval(M,[epsilon=1/100,d=3,m=1,p=1])); # Exact integration
evalc(%)=evalf(%);
evalf(eval(M,[epsilon=1/100,d=3,m=1,p=1])); # Numerical integration
```

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