I never heard of the method of isodine. May be you meant isoclines  ?

To make slope plot do

ode:= x+y(x)*diff(y(x),x)=0;
DEtools:-DEplot(ode,y(x),x=-2..2,y=-3..3)

To get the solution, the command is 

dsolve(ode)

To get step by step solution the command is

Student:-ODEs:-ODESteps(ode);

evalb(expand(sin(x + y))= sin(x)*cos(y) + cos(x)*sin(y)); 

    true

Maple does not expand by default. (which is a good thing)

Can you recommend a good way to find out if two terms are equivalent?

There are few ways. One easy way is to do simplify(A-B) and see if you get zero

simplify(sin(x + y) - (sin(x)*cos(y) + cos(x)*sin(y)))
 
      0

But ofcourse this depends how good the simplify command is. Also you can help the simplify with assumptions if needed.

There is also the command is and coulditbe to try

is(sin(x + y) = sin(x)*cos(y) + cos(x)*sin(y))

              true

There might be other ways.  I had few false positives from coulditbe so be careful with it.

worksheet below generates this pdf

THis is latex generated

\documentclass[12pt,a4paper]{article}
\usepackage[letterpaper,margin=1.2in]{geometry}
\usepackage{enumitem}
\begin{document}
\begin{enumerate}[label=\arabic*)]
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$ \quad $C(-10; -2; 3)$, \quad $(P) :10 x +6 y +z +109 = 0$
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$ \quad $C(-10; 6; 3)$, \quad $(P) :2 x -2 y +z +29 = 0$
\item $A(-12; 2; -1)$,\quad $B(-11; 1; -5)$ \quad $C(-9; 5; -7)$, \quad $(P) :3 x -y +z +39 = 0$
\end{enumerate}
\end{document}

Download convert_list_to_latex_may_24_2024.mw

I am no expert on this. I find that using pattern matching in CAS to be more natural than using structured types to do this sort of thing. It also does not help that Maple documentation of its most important thing, which is structured typing, has so few examples to learn from. May be if Maple help had extensive examples, doing this sort of thing will not be like black magic any more to new users.

But here is an attempt. I am sure there is better way to do this in Maple.

Download parsing.mw

look at evalc

expr:= exp(alpha[i]*I*t);
evalc(expr)

one way

sol := (-v + sqrt(-4*a^2*R^2 + v^2))/(2*a*omega*L);
expand(sol);
map(X->`if`( hastype(X,'anything'^(1/2)),sqrt(X^2),X),%)

Another using pattern matching

sol := (-v + sqrt(-4*a^2*R^2 + v^2))/(2*a*omega*L);
f:=proc(x)
local a,b,c,la;
if patmatch(x,b::nonunit(anything)*sqrt(a::nonunit(anything)),'la') then
   assign(la);
   RETURN(sqrt(a*b^2));
else
   RETURN(x);
fi;
end proc;
map(X->f(X),expand(sol));

note that in all the above,  sqrt(a)/b is same as sqrt(a/b^2) assuming b>0

Yes, this happens, but the important thing is that the corresponding order with the correct eigenvalue do not change.

i.e the way to read the outout is that the first eigenvalue goes with the first column, the second eigenvalue goes with the second column and so on.

So it does not matter if the eigenvector columns change positions, as long the the corresponding eigenvalues change in same way

btw, if you for some reason need to have same order of eigenvectors each time, you could always sort the eigenvector matrix columns using the numerical values of the corresponding eigenvalues as key for sorting. Something like

Download sorted_V.mw

Maple 2024

I have feeling there is a better or more direct way to do this but could not find it so far. But here is an attempt

Opps. Fix to make true matrix

R:=Vector[row]([1,2,3]);
C:=Vector([d,e,f]);
convert(convert(map(X->X*C,R),listlist),Matrix);
whattype(%);
LinearAlgebra:-Dimension(%%);

You should really post plain text code. One can't copy code from image. But you can try

sol:=signum(-sigma+q)^2;
simplify(sol) assuming real;

You should really post plain text code next time or at least worksheet.

I tried this in Maple 2024 and Mathematica 14. Both give solution that validate.  Here is Maple's

ode1:= diff(c(T),T)=-2*c(T)*(1+beta__c*c(T)^2/p__c(T)^2);
ode2:= diff(p__c(T),T)=2*(1+beta__c*c(T)^2/p__c(T)^2)*p__c(T);
sol:=[dsolve([ode1,ode2],[c(T),p__c(T)],'explicit')]

Download it_works_for_me.mw

Here is Mathematica code I tried.

ClearAll["Global`*"]
ode1 = c'[T] == -2*c[T]*(1 + \[Beta]*c[T]^2/pc[T]^2);
ode2 = pc'[T] == 2*(1 + \[Beta]*c[T]^2/pc[T]^2)*pc[T];
sol = DSolve[{ode1, ode2}, {c, pc}, T]

Solution also verifies OK

Since both solutions from Maple and Mathematica verfiy OK, then both are correct even though they look different. 

set has no implied ordering. So need to use list.  Here is one way out of many

X:=[x12, x1, x3, x15, x2, x9];

f:=proc(a,b)::truefalse;
     :-parse(String(a)[2..]) < :-parse(String(b)[2..]) ;
end proc:

sort(X,f);

may be

n:=294912:
d:=8:
eq:=d^m=n/2:
floor(solve(eq,m)):
n/(d^%);

        9

This assume n is even 

You can easily modify it for odd n 

Need to tell is x is not negative

simplify(g(f(x))) assuming x>=0

                 x

This is a guess. When expression is product, such as a*b internally it is   a^1*b^1  so  when you do subs 1=2 you get a^2*b^2. It replaced the exponents by 2.

same for division. You can see this from

subs(1=2,a*b^9)

           a^2*b^9

And 

dismantle(a*b)

PROD(5)
   NAME(4): a
   INTPOS(2): 1  ------->
   NAME(4): b
   INTPOS(2): 1  ----->

Compare to 

dismantle(a*b^9)

PROD(5)
   NAME(4): a
   INTPOS(2): 1   --->
   NAME(4): b
   INTPOS(2): 9   ---->

So internally for a  PROD   , each term has hidden exponent of 1 by default unless overriden.

For SUM, the same thing. There is a hidden or implied  `1`. You can see this from

dismantle( a + b);

SUM(5)
   NAME(4): a
   INTPOS(2): 1 --->
   NAME(4): b
   INTPOS(2): 1  --->

Compare to 

dismantle( a + 9*b);

SUM(5)
   NAME(4): a
   INTPOS(2): 1  ------->
   NAME(4): b
   INTPOS(2): 9  ------->1

So when you do 

subs(1=2,a+b)

It is as if  you typed

subs(1=2, 1*a+ 1*b)

And that is why you get 

          2*a + 2*b

It is by design. This is called "infinite scroll"

Google search also introduced this not long ago on its search result. It is terrible feature and Maplesoft seems to copy this. 

Google Infinite Scroll keeps loading new results as you scroll down the page.

Many web sites seem to imitate this now. 

I much prefer the old way. One page at a time, and click next page to see the next page of result.

I could not find a way to turn it off from google search result.

For Maple's page above, you'd have to ask Maplesoft if they can turn off Infinite Scroll as it is part of the page design and they are the ones who add the javascript in the page to control this.

Outside users has no control as far as I can see on how to prevent Infinite Scroll

So what you see as repositions arbitrarily. is actually a side effect of this automatic scrolling. If not done right, it looks like what you describe and I've seen this in many other sites.

It also makes one lose track of what they were looking on the page when this happens.

The web is a big mess and Maplesoft is just following the trend.