I have seen other cases where it pays to consider a more general version.
Anyway, here are two animations in b. The first shows that of the 2 solutions found only the second solves the equation on some interval.
 

restart;
sol:=int(1/sqrt(sin(y)),y);
solve(sol=x+b,y):
res:={%};
numelems(res); #2
res1b:=simplify(eval(sol-x-b,y=res[1]));
res2b:=simplify(eval(sol-x-b,y=res[2]));
plots:-animate(plot,[res1b,x=-10..10],b=-5..5,trace=24);
plots:-animate(plot,[res2b,x=-10..10],b=-5..5,trace=24);

Here is an image of the end of animation 2:

@charlie_fcl I only had to fight to get rid of the .txt in maple.ini.txt.

When I did your worksheets ran.

@charlie_fcl Yes, but there are significant changes made in Maple 2024.

Is rtable_alias documented anywhere?

@Mathrookie94 You only have to do this:

DEtools:-DEplot(ODE,y(t),t=0..350,[y(0)=449],y=0..2000);

where ODE is:

ODE := diff(y(t), t) = -0.000032229*y(t)^2 + 0.065843*y(t) - 15.103;

Obviously, if you have another ode you will want to change the numbers in the first line.
Thus t=0..350, [y(0)=449], and y=0..2000 may be different.
If the dependent variable isn't y(t), but say x(s), you will have to change t to s and y to x.
Here is an example:

restart;
ode:=diff(x(s),s) = -(1 - 2*x(s))*(1 - x(s))*x(s);
#There are 3 constant solutions: x(s) = 0, x(s)=1/2, and x(s) = 1.
#They make the right hand side of ode equal to zero.
#The 3 constant solution and 4 other solutions are plotted here:
DEtools:-DEplot(ode,x(s),s=0..8,
   [x(0)=0,x(0)=1/2,x(0)=1,x(0)=-1/2,x(0)=1/4,x(0)=3/4,x(0)=3/2],x=-1/2..3/2);

Since you are able to use pointplot right after restart, i.e. you don't need plots:-pointplot, you must have a maple.ini file or some start up code.

So what is in that?

@mmcdara I agree.  In fact why should Maple do anything to 1h and 5 min. I myself like that better than 65 min.

@nm Very nice. I would, however, add one requirement:
 

if not has(eq,diff) or not T::`+` or not has(eq,yx) then return eq end if;

Your test case number 2 isn't an ode in y(x) so ought not be handled as such in my view.

The same can be written:

if not ( has(eq,diff) and T::`+` and has(eq,yx) ) then return eq end if;

@nm Number 3 and number 9 work as expected since the very first line in q:

if not has(eq,diff) then return eq end if;

makes those return unevaluated.
Number 7, however, certainly shows a weakness that should be handled.

@nm Thanks for making these points.
The first corrects a silly mistake on my part.
The second point is more tricky.
Here is an attempt:

p:=proc(eq::equation,{known::function:=NULL}) local d,fu,res;
  if not has(eq,diff) then return eq end if;
  d:=indets(indets(eq,specfunc(diff)),function) minus {known};
  fu:=op(remove(has,d,diff));
  if numelems({fu})>1 and known=NULL then 
    error "%1 has derivatives of more than one function. Use a second argument: known=function",eq end if;
  res:=selectremove(has,(lhs-rhs)(eq),fu);
  res[1]=-res[2]  
end proc:
########################
pde:=diff(u(x, t), t, t) + 3 + 2*diff(u(x, t), t) + 4*t + x^2 + x^3/3 + diff(u(x, t), t, x, x) + diff(u(x, t), x, x, x, x) = x*t^2;
ode:=y(x)+diff(y(x),x$2)+x=sin(x);
ode1:=y(x)+diff(y(x),x$2)+diff(f(x),x)=sin(x);
########
p(ode);
p(pde);
p(ode1);
p(ode1,known=f(x));

A much easier solution is to require a second argument, the function to solve for.

q:=proc(eq::equation,yx::function) local res;
  if not has(eq,diff) then return eq end if;
  res:=selectremove(has,(lhs-rhs)(eq),yx);
  res[1]=-res[2]  
end proc:
q(ode,y(x));
q(ode1,y(x));
q(pde,u(x,t));

For the fun of it I changed the ode's right hand side to 1/sqrt(Pi):

ode1:=diff(y(x),x)+x*y(x)=1/sqrt(Pi);
dsolve(ode1); # No Pi
ic:=y(0)=0;
Student:-ODEs:-ODESteps([ode1,ic])

That works fine.

@Carl Love Thanks Carl. I will use that too from now on.

@acer Thank you acer.

@acer I use that kind of thing in my maple.ini file , but in a version using sscanf called to my attention by Joe Riel a few years ago.
https://www.mapleprimes.com/maplesoftblog/209424-The-Maple-Command-Line-Interface-Now-In-Color

I need the digit after the period.
So I would use:
 

restart;
str:=convert(kernelopts(version),string);
### Also parse: 
N:=parse(str[7 .. StringTools:-Search(".",str)+1]); # i.e. +1 instead of -1

This makes it possible in my maple.ini file to distinguish between versions as in
if N> 2023.2 then .... end if.

So I have been using this:
 

restart;
VERSION:=kernelopts('version');
   N:=op(sscanf(VERSION, "Maple %f[0-9]"));  
   forget(evalf); #Otherwise the remember table will have an entry like 2019.2 = (2019.2, 10)
# I don't quite see the need for forget anymore.

@nm Try this:

restart;

sol:=y(x) = (exp(RootOf(-sin(x)*tanh(1/2*_Z+1/2*c__1)^2+sin(x)+exp(_Z)))+sin(x))/sin(x);
ode:=diff(y(x),x)-cot(x)*(y(x)^(1/2)-y(x)) = 0;

sol1,sol2:=allvalues(sol);

sol1:=simplify(expand(sol1));
sol2:=simplify(expand(sol2));
sol1_0:=eval(sol1,c__1=0);
sol2_0:=eval(sol2,c__1=0);
eval(rhs(sol1_0),x=-Pi/2); # 0
eval(rhs(sol2_0),x=-Pi/2); # 4
dsolve({ode,y(-Pi/2)=0});# y(x) = 0
solDs2:=dsolve({ode,y(-Pi/2)=4});
solDs2:=simplify(value(solDs2)) assuming x<0,x>-Pi;
plot(rhs(solDs2)-rhs(sol2_0),x=-Pi..0,style=point); # basically zero
plot(rhs(solDs2),x=-Pi..0,0..15,color=red); p1:=%: 
plot(rhs~([sol1_0,sol2_0]),x=-Pi..0,0..15); p2:=%:

plots:-display(Array([p1,p2]));

Interestingly you find two solution just as dsolve does, but one of the dsolve solutions is just the trivial y(x)=0.
Both yours and dsolve's can be correct since the initial value problem with y(-Pi/2) = 0 doesn't satisfy the requirements for uniqueness.
Note added:
Your sol1_0 isn't in fact a solution to the ode in the interval I consider here.
The value of sol1_0 at x=-Pi/4 is 1 - 2*2^(1/4) + sqrt(2) which is positive.
Using dsolve with that initial condition gives a solution that is valid on the interval
-27*Pi/32 ..-5*Pi/32.
On that interval sol1_0 and the dsolve solution differ:

@jestrup Sorry, now I get it. 
By "This could be relevant when using Maple for a test"  you mean that you don't want students to be able to access the AI formula assistant during a test.
As long as the students have a copy of Maple on their computers, I don't see any way you can do that except maybe for turning off your wi-fi.