sometimes I get equations where there is clearly more simplification on it. For an example

restart;
ode:=y(x)=diff(y(x),x)^3*y(x)^2+2*x*diff(y(x),x);
new_ode:=PDEtools:-dchange({y(x)=sqrt(u(x))},ode,{u});

Doing this by hand, I would now write the above as

by multiplying both sides by sqrt(u(x))

new_ode :=sqrt(u(x))*lhs(new_ode)=simplify(sqrt(u(x))*rhs(new_ode))

Since I am doing this in a program, without knowing what the expression is, I need to have this simplification done automatically, since I do not know before hand, if the resulting expression has this form or not. So I automatically call simplify on it.

I tried on a simpler example

expr:=sqrt(u)=1/sqrt(u)

And want to simplify this to u=1

I can't do  

new_expr :=sqrt(u)*lhs(expr)=sqrt(u)*rhs(expr)

Since this is done in a program, without the benefit of looking first and then deciding what to do. Everything needs to be automated since the expression can be anything.

So I need a generic operation to apply and have Maple simplify it, if this pattern exist.  I tried many things, but do far nothing is working:

restart;
expr:=sqrt(u)=1/sqrt(u);
expand(expr);
expand(expr) assuming real;
combine(expr);
combine(expr,sqrt);
combine(expr,radical);
combine(expr,power);
expand(expr);
collect(expr,sqrt(u));
collect(lhs(expr)-rhs(expr)=0,sqrt(u));
simplify(lhs(expr)-rhs(expr)=0) assuming u<>0; # I expected this to work
simplify(expr) assuming real;
simplify(expr,size=false);
simplify(expr,sqrt) assuming real;
simplify(expr,sqrt) assuming positive;
simplify(expr,sqrt,symbolic) assuming positive;
simplify(expr,sqrt,symbolic,size=false) assuming positive;
simplify(expr,symbolic,size=false);
simplify(expr) assuming u<>0;
rationalize(expr);

no change.

Please do not give me an answer that requires one to use specific solution assuming one can see the expression. The solution needs to be something generic and work on any such expression that I can apply to any expression and have Maple simplify the sqrts on both sides, if they are there, since this is done in a program.

From the above, this should have worked

expr:=sqrt(u)=1/sqrt(u);
simplify(lhs(expr)-rhs(expr)=0) assuming u<>0;
simplify(lhs(expr)-rhs(expr)=0,symbolic);

Using Mathematica

ClearAll[u];
expr = Sqrt[u] == 1/Sqrt[u]
Simplify[expr[[1]] - expr[[2]] == 0, Assumptions -> u != 0]

Any suggestions? I am sure there is a simple way to do this in Maple, I just have not found it yet.

Maple 2020.2

Why Maple can't simplify this expression to zero?

restart;
ode:=diff(y(x),x)-y(x) = x*y(x)^(1/2);
ic:=y(0)=4;
sol:=dsolve([ode, ic],y(x));
check:=odetest(sol,ode);

simplify(check) assuming x>0

In Mathematica:

ClearAll[x]
check = -2*x + 4*x*Exp[x/2] - x^2 -x*Sqrt[x^2 - 8*x*Exp[x/2] + 4*x + 16*Exp[x] - 16*Exp[x/2] + 4]
Simplify[check, Assumptions -> x > 0]

note that using x>=0 instead of x>0 does not change the above output.

Any work around in Maple?

I think this is also wrong ode type given by ode advisor.   The following clearly can't be _dAlembert ode. 

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

It does not even have an explicit x on its own, in the RHS. D'Alembert ode has the form

The ode above has the form   y=f(p)+g(p)  or one can argue the form y=F(p). Either way, there is no x in the RHS.

For the first form above, f(p)=p*ln(p) and g(p)=ln(cos(p))

So why Maple says

DEtools:-odeadvisor(ode)

And Maple knows this. But for some reason, for the above ode, I think it made the wrong call.

restart;
ode:=y(x)=x*f(diff(y(x),x))+g(diff(y(x),x));
DEtools:-odeadvisor(ode);

ode:=y(x)=f(diff(y(x),x))+g(diff(y(x),x));   #no x
DEtools:-odeadvisor(ode)

Is there something I am overlooking here? 

Maple 2020.2 with Physics 908 on windows 10

I need to find and remove any abs that shows up only inside ln anywhere in an expression.

I used to do this in multiple steps before, by using loop and subs. But I am now learning evalindet which is powerful command. I should use it more. 

I wanted to see if it is possible to do this in one call to evalindent. Both finding and replacing.

Here is an example. Given

expr:=sin(x)+ln(abs(x))+ln(x+abs(y)/sqrt(abs(x+3)))+ln(x^3)+cos(abs(x));

The goal is to change it to the following

This is what I ended up with

restart;
expr:=sin(x)+ln(abs(x))+ln(x+abs(y)/sqrt(abs(x+3)))+ln(x^3)+cos(abs(x));
expr:=evalindets(expr,
          'specfunc( satisfies(u->has(u,abs)),  ln)',
          f->evalindets(f,'specfunc(anything,abs)',f->op(1,f))
          )

My question is: Is there a more optimal or better way to do this? I had to use evalindets inside the transformer to remove the abs. At first I did not know if it will work, but it did work.

Any place for improvement?

Maple 2020.2

Given an expression, I want to obtain list of all functions in it, that contains as argument, anywhere another function.  

I can do this now using 2 steps. I was wondering if there is a way to do it in one call.

Here is an example.

expr:=sin(x)+ln(abs(x))+ln(x+1/sqrt(abs(x+3)))+ln(x^3);

The goal is to find all ln functions, with abs inside them. Now, I do this

restart;
expr:=sin(x)+ln(abs(x))+ln(x+1/sqrt(abs(x+3)))+ln(x^3);
lis:=indets(expr,'specfunc(anything,ln)');
select(Z->has(Z,abs),lis)

I could not find a way to do it in one call, If I do this

indets(expr,'specfunc(abs(anything),ln)');

it only finds

Which overlooked the other one, since abs there is elsewhere in side ln.

I looked at help Definition of a Structured Type in Maple but do not know yet if there is an option there to do this. 

Is it possible to do this on one call?  Just wondering, that is all. The above works OK for me now.

Maple 2020.2