I was checking my solution against Maple. Maple gives solution with integral and RootOf. I am not able to simplify it to better compare.

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode);

It is the second solution above I want to simplify/evaluate. So I tried

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode);
sol:=[sol][2];
DEtools:-remove_RootOf(sol)

But this did nothing. Then I tried adding useint in the dsolve command

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode,useint);

But this also kept the integral there. Then I tried eval

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=[dsolve(ode,'explicit')];
eval(sol[2])

The solution I obtained is 

y(x) = -ln(_C1^2 - 2*_C1*x + x^2 + 1)/2 + ((-2*x + 2*_C1)*arctan(-x + _C1))/2

Which does verify to zero OK using odetest.

Any suggestions/tricks to use to get an more explicit solution from Maple I am overlooking?

Maple 2020.1 , Physics 724

I can post/attach my full solution if needed. 

I solved this ODE and got a solution and wanted to compre it with Maple. This is initial value first oder ODE. So it should have no constants in it. But Maple's solution contains something I never seen before _B1~

I wonder what it means? And odetest did not verify Maple's solution. 

restart;
ode:=diff(y(x),x)-y(x)/x+csc(y(x)/x)=0;
sol:=dsolve([ode,y(1)=0]);
simplify(odetest(sol,ode));

odetest does not gives zero.

This is my solution

mysol:=y(x)=x*arccos(ln(x)+1);
odetest(mysol,ode)

   0

Any idea what _B1~ means? The ~ looks like it is an assumed variable? may be leaked from inside Maple.

Maple 2020.1 on windows 10

I use odeadvisor a lot to tell me the type of the ode. It is one of the best tools in Maple.

But sometimes it overlooks some types of ODE's, if the ODE is written in different way. This does not happen alot. Here is an example

ode:=diff(y(x),x) = (2*x+y(x))/(3-x+3*y(x)^2);
DEtools:-odeadvisor(ode)

And advisor says it is rational, which is correct. 

But it does not also say it is exact. By rewriting as follows, it now see it is exact as well as rational:

ode2:=(denom(rhs(ode)))*diff(y(x),x)-(numer(rhs(ode)))=0;
DEtools:-odeadvisor(ode2);

And now it says [_exact, _rational]. It is the same ODE, just written different.

This is not a complaint about the advisor, I know it is not easy to figure the type of the ODE under different trasformations, but may be something to look into to improve it to be able to detect more types.

Maple 2020.1 

One thing I always liked about Maple, is that it does not do automatic simplifications or automatic re-write of expressions without the user explicitly asking for it like some other CAS systems do, which I think this is the right way.

So I was surprised when I set up a rational expression of a numerator and denominator, then when I asked for its denom and numer, I found Maple did automatic rewrite (even though on the screen the expression remained as it was originally). This caused a bug in my program (since it assumed as long as no simplifcation is made on the expression, numer and denom will remain as it was orginally placed). Here is an example

restart;
num:=-(1-x/exp(y));
den:=(exp(y)+x);
expr:=num/den;

So now, one would expect, by looking at the screen, when asking for numer(expr) to obtain back -(1-x/exp(y))  but this is not what happens

numer(expr)

And 

denom(expr)

So Maple's internal representation of numer and denom, is not what "appears" on the screen for the user. Maple automatically replaced -1+x/exp(y) by (-exp(y)+x)/exp(y) and then moved denominator of this, which is exp(y), down to the denominator of the orginal expression.

I was surprised by this, since as I said, my experence with Maple is that it keeps expressions as entered and will change them only when the user asks for a change.

How common is such behavior in Maple? and why did Maple do this under the cover manipulation in this case? It is probably documented somewhere if I search hard enough.

I was trying to compare my solution with Maple on this ode, when I noticed Maple gives solution with no y(x)= in it. 

restart;
ode:=(y(x)-x*diff(y(x),x))/(y(x)^2+diff(y(x),x))=(y(x)-x*diff(y(x),x))/(1+x^2*diff(y(x),x));
sol:=dsolve(ode)

When there is more than one solution, Maple write y(x) on each. For an example

dsolve(diff(y(x),x)^2=x)

How to explain this output?

Maple 2020.1