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  • $\begingroup$ These DEs are not separable. They can be transformed to separable ones. It should be possible to separate separable ones. I think finding transformations is difficult. DSolve probably does it sometimes, although it might choose a different solution method in other cases. $\endgroup$ Commented Aug 16, 2024 at 13:47
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    $\begingroup$ For instance, DSolve`DSolveFirstOrderODEDump`SeparateVariables[(1 + x - y)/(-3 + x + y), x, y] returns failed, but DSolve`DSolveFirstOrderODEDump`SeparateVariables[(1 + x - y)/(-3 + x + y) /. {x -> (x + y)/Sqrt[2], y -> (x - y)/Sqrt[2]}, x, y] succeeds. $\endgroup$ Commented Aug 16, 2024 at 13:59
  • $\begingroup$ see how-to-determine-if-rhs-of-first-order-ode-is-separable-or-linear-or-neither-pa and Leonid Shifrin answer how-can-i-separate-a-separable-function But I found some cases where these do not detect. So I have added 60 more patterns in my own ode solver as extra checks. But the above should do it for most common separable ode's. btw, your odes are not separable. $\endgroup$ Commented Aug 16, 2024 at 15:43
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    $\begingroup$ automatically separates variables for separable ODE? may be there is different convention you are using for separable ode. For me, a separable ode is $y'=F(x,y)$ which can be written as $y'=f(x) g(y)$ and the goal then is to find if $F(x,y)=f(x) g(y)$ and ofcourse to find $f(x),g(y)$. The above linked to answer does that. But as I said, there are few edge cases it might not work for. But for most common odes', it works fine. $\endgroup$ Commented Aug 16, 2024 at 16:55
  • $\begingroup$ Thank you for your very valuable comments, and I ' ll take the time to study what you mentioned. For separable ODE, what I describe is not accurate enough and has been updated. $\endgroup$ Commented Aug 16, 2024 at 23:27