Symbolics.jl can’t solve this equation but if I multiply both sides by x^3 then it can. Wolfram Alpha can solve both. Symbolics.jl’s docs say it can do polynomial equations; do these count? What’s the problem?

[2edaba10] Nemo v0.46.2
[0c5d862f] Symbolics v6.11.0
using Symbolics, Nemo
@variables x y
julia> symbolic_solve(1/x^2 ~ 1/y^2 - 2/x^3 * (x-y), x)
┌ Warning: This expression cannot be solved with the methods available to ia_solve. Try a numerical method instead.
└ @ Symbolics ~/.julia/packages/Symbolics/6WqId/src/solver/ia_main.jl:176
julia> symbolic_solve( (1/x^2 * x^3 ~ (1/y^2 - 2/x^3 * (x-y)) * x^3), x)
[ Info: Assuming (y^2) != 0
[ Info: Assuming (y^2) != 0
2-element Vector{SymbolicUtils.BasicSymbolic{Real}}:
y
-2y

Nemo is needed to enable a package extension in Symbolics. The problem is that only polynomial equations can be solved, so you need to multiply out any denominators. Here’s a quick way to reproduce the problem.

julia> using Symbolics, Nemo
julia> @variables x;
julia> symbolic_solve(x ~ 1/x, x) # denominators cannot be handled
┌ Warning: This expression cannot be solved with the methods available to ia_solve. Try a numerical method instead.
└ @ Symbolics ~/.julia/packages/Symbolics/e7UFe/src/solver/ia_main.jl:176
julia> symbolic_solve(x^2 ~ 1, x) # multiply both sides by `x` to get polynomials
2-element Vector{BigInt}:
-1
1

It should be fixed in Symbolics, since Nemo is used as a backend. But since their recent feature announcement mentions “multivariate polynomial solving”, I understand that is a feature limitation not a bug.

There isn’t a reason this cannot be handled via some normalization pass that occurs before the solve and then fixes the solution after the solve. Definitely worth an issue