I was wondering why dsolve do not show this one solution I obtained different way, for this ode.

Maple gives 6 solutions to the ODE. But 5 of them are signular (have no constant of integration in them). I am looking at the last solution it gives, the one with constant of integration. 

But when I solve this ODE, after simplifying it, I get different general solution than the last one Maple shows. I thought they might be equivalent, but I do not see how they could be. 

So my question is why Maple did not show the simpler solution also?  Here is the code

ode_orginal:=1/3*(-2*x^(5/2)+3*y(x)^(5/3))/x^(3/2)*diff(y(x),x)/y(x)^(5/3)+1/2*(2*x^(5/2)-3*y(x)^(5/3))/x^(5/2)/y(x)^(2/3) = 0;

This looks complicated and non-linear, but when I solve for y'(x) it gives much simpler and linear ODE (Have to convert the ODE to D before solving for diff(y(x),x), since Maple can complain otherwise).

ode:=convert(ode_orginal,D);
ode:=diff(y(x),x)=solve(ode,D(y)(x),allsolutions=true);

And solving the above gives the much simpler solution

sol:=dsolve(ode,y(x));
odetest(sol,ode);

Now when using dsolve on the original complicated lookin ODE it gives

maple_sol:=[dsolve(ode_orginal)];

if there is any hope that one of the above 6 solutions will be the same as the simpler solution, it has to be the last one, with the _C1 in it, since all the others do not have _C1 (singular solutions).

But solving for y(x) from the last one does not give the simpler solution.

So it is a new general solution for the original ode and it is correct, since odetest gives zero.

I think in the process of solving for y'(x), some solutions got lost. Even though I asked for allsolutions there?.

But my main question is: Should dsolve have also have given the simpler solution? Since that one also satisfies the original nonlinear complicated ode

odetest(sol,ode_orginal);

gives 0.

Maple 2021.1

I solved this ode with IC, and obtained a solution.  Maple odetest says my solution satisfies the ODE but not the initial conditions. But I do not see why. When I plugin manually the initial conditions into the solution, I get true.

So I am not sure what is going on.

restart;
ode:=x*diff(y(x),x)*y(x) = (x+1)*(y(x)+1);
ic:=y(1)=1;
mysol:=y(x)-ln(y(x)+1)=-ln(2)+x+ln(x);
my_sol_at_IC:=subs([y(x)=1,x=1],mysol);

#now check manually
evalb(my_sol_at_IC);
is(my_sol_at_IC);

Using odetest gives

odetest(mysol,ode);
odetest(mysol,[ode,ic]);

What Am I doing wrong? 

Maple 2021.1

Edit June 12, 2021

This is another example where odetest gives different result depending on how the solution is written !

restart;
ode:=diff(y(x),x)=2/3*(y(x)-1)^(1/2):
mysol_1:=y(x) = 1/9*x^2 - 2/9*x + 10/9:
mysol_2:=y(x) - (1/9*x^2 - 2/9*x + 10/9) = 0 :
odetest(mysol_1,ode);
odetest(mysol_2,ode);

I can't figure what is going on here. I have an expression, where when I run the program, I see inside the debugger that the result of evalb(simplify(tmp_1)) where tmp_1 is the expression, gives false

When I copy the same tmp_1 expression to new worksheet and do evalb(simplify(tmp_1)) it gives true.

I added print statements in the code to print tmp_1 and print result of simplify(tmp_1) and for some reason, it gives false when I run the program, but it gives true when I do the same thing in a work sheet, by copying tmp_1 and trying it there.

Never seen anything like this, as expression is all numbers. Here is the print out when I run the program

Here is the same thing in worksheet

restart;
tmp_1 :=-(2*ln(-2))/7 - ln(-1)/7 = ln(1) - (2*ln(-2))/7 - ln(-1)/7;
simplify(tmp_1);
evalb(simplify(tmp_1));

                      true

and if you think I made mistake somewhere, here is also screen shot from the debugger window itself

again, same thing gives true in separate worksheet, copying same exact expression from above

There seem to be something loaded when I run my program that causes this difference, but I have no idea now what it is.  This has nothing to do with the debugger loaded btw, I am just using the debugger to show the details. When I run the program, without the debugger, I get false for the above, and I was trying to find why.

Any suggestions what to try and what to check for and what could cause this? Should it not give true when I run the program also? Could it be some Digits setting changed when the program run? I do not change any system defaults of any sort when running the program. 

Maple 2020.1

edit june 12, 2021

Here is another manifestation of the same problem I just found. There is something seriously wrong for it to behave this way.

This is an expression called tmp_1 in a local variable in a proc inside a module. There are not even any local symbols in it. When I run the program, all the attempts to show it is true fail. Some give FAIL and some give false when the result should be true

When I simply copy the expression from the debugger window to an open worksheet, I get true

Here is the screen shot

Only when I did is(expand(tmp_1)) did it give true

But when I copy the expression to the worksheet, it gives true using all the other attempts:

restart;
tmp_1:=1 = 5/4*exp(0)+1/4*piecewise(0 <= 1,-1,1 < 0,exp(-2)):
is(tmp_1);
is(simplify(tmp_1));
is((rhs-lhs)(tmp_1)=0);
is(simplify((rhs-lhs)(tmp_1)=0));
evalb(simplify((rhs-lhs)(tmp_1)=0));
is(expand(tmp_1))

I can't make a MWE so far, since these problems only show up with I run my large program. I think Maple internal memory get messed up or something else is loaded that causes these stranges problems.

For now, I added expand() to the things I should try. My current code now looks like

if is(tmp_1) or is(simplify(tmp_1)) or is((rhs-lhs)(tmp_1)=0)
               or is(simplify((rhs-lhs)(tmp_1)=0)) or evalb(simplify((rhs-lhs)(tmp_1)=0))
               or is(expand(tmp_1)) then    
   ....

Maple 2021.1

I need to change any occurance of   anything*sqrt(anything)  to say Z in any large expression.  (later, I can add the correct replacement once I know how to do it for Z).

I can change   sqrt(anything) with the help of the answers in Substitution-Of-Sub-Expressions-Is  but not  anything*sqrt(anything)

Here is an example to make it clear. Given

expr:=(-x + sqrt(9*x^2*exp(2*c) + 8)*exp(-c)+99)/(4*x)+ (a*sqrt(z)-99+sin(c*sqrt(r+4)+20))/3+10+1/(c+exp(-x)*sqrt(exp(x)))+sqrt(h)

Need to all some transformation on  those subexpressions circled above. For now, lets say I wanted to replace them with Z so it should becomes this

I can do this

subsindets(expr,anything^({1/2,-1/2}),ee->Z)

But I need to have the term (if any) that multiplies the sqrt as well included.

And that is the problem. Can't figure how to do it. When I try

subsindets(expr,anything*anything^({1/2,-1/2}),ee->Z)

Maple says Error, testing against an invalid type Ok. So anything*anything^({1/2,-1/2}) is not a type. What to do then?  Tried also subsindets(expr,t::anything*anything^({1/2,-1/2}),ee->Z) same error

And 

applyrule(t1::anything*sqrt(t0::anything)=Z, expr);

gives  Z which is wrong.

How to make this work in Maple using subsindets? Can one use pattern with subsindets? How to make a type for

               anything*sqrt(anything) 

Maple 2021.1

Why Maple likes to extract exp() outside the sqrt when its argument has minus sign vs. not?  Compare the following

restart;
eq2 := ln(2*u^2 + u - 1) = -c - 2*ln(x);
sol:=[solve(eq2,u)]:
simplify(sol[1])

and

eq2 := ln(2*u^2 + u - 1) = c - 2*ln(x);
sol:=[solve(eq2,u)]:
simplify(sol[1])

I like the above much better than the first one. Mathematica keeps both same form (i.e. keeps the exp() inside):

Maple's answers are correct ofcourse, I just do not understand the logic why when there is a minus sign on c it likes to format it differently as shown.

Is there a way to make not do that?