I check the help center and find the following commands:

PDETools:-PolynomialSolutions This command catch explicit polynomial solution y=f(x)  where f  is polynomial.

DETools:-ratsols  returns explicit rational solutions for linear differential equation.

But I don't seem to find any commands to compute implicit algebraic curve solution f(x,y)=0  for, say for instance, non-linear first order ODE with derivative solved explicitly.

@dharr 

Thanks for your advice

@dharr 

It works for this case. But can you try this one? Solve each variables in terms of parameters "x"

solve_test_(1).mw 

@dharr 

Thank you so much! Really appreciate your code!!

@dharr 

But can I ask for a further question? How can I know which variables are to be chosen to be variables to solve for and which are free variables? For example in this case, if I choose b[0],b[1],b[2] to be variables to solve for, I would get an empty solution set.

@Carl Love But I indeed look for a six variables system. Three equations for six variables, t is treated as constant. It's an overdetermined system.

Mistake in (71) tanh(arctanh(y)/2)

If the ODE can be solved directly, the polysols first calculate the general solutions and find poly solutions in the general solutions so the singular solution is ignored. If the ODE cannot be solved directly, the polysols use some algorithm to find some particular singular solutions.

the parameters a=c,b=d

@Carl Love Thank you!

Please just ignore this question