remove bug and make it more robust

[RituRajSingh878]

References to other Issues or PRs

Brief description of what is fixed or changed

During separation,separatesvarscheks each multiplicative factors of the equation and find out that the factor is separable or not and it seems good. But if we will send an equation with expand(eq) so all the factor of equation will destroy and now separatevars will may not able to separate the equation. So I have added factor(eq) so that we can make it a multiplicative equation if possible.

And second, think is a bug- passing of force into separatevars in the function _separatevars.

And the third thing is after getting a multiplicative equation when we are passing each factor again in separatevars then maybe after doing simplification of each factor we can get a better result.
I tried this and able to separate the more equations. (see added test case)

eg-
eq=exp(x**2+y**2)*(cos(x-y)+cos(x+y))

after passing this term we are getting one of the factors - (cos(x-y)+cos(x+y)) and passing this only in separatevars we are not able to separate it so I think we can pass simplify(factor) every time in separatevars so that we can get a simplified equation and then we can easily separate the equation.

Other comments

Release Notes

• simplify
  • remove a bug from separatevars and make it more robust

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• simplify
  • remove a bug from separatevars and make it more robust (#17442 by @RituRajSingh878)

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#### Brief description of what is fixed or changed
During separation,` separatesvars `cheks each multiplicative factors of the equation and find out that the factor is separable or not and it seems good. But if we will send an equation with expand(eq) so all the factor of equation will destroy and now `separatevars` will may not able to separate the equation. So I have added factor(eq) so that we can make it a multiplicative equation if possible.

And second, think is a bug- passing of force into `separatevars `in the function `_separatevars.`

And the third thing is after getting a multiplicative equation when we are passing each factor again in `separatevars` then maybe after doing simplification of each factor we can get a better result.
I tried this and able to separate the more equations. (see added test case)

eg-
 `eq=exp(x**2+y**2)*(cos(x-y)+cos(x+y)) ` 

after passing this term we are getting one of the factors -  `(cos(x-y)+cos(x+y))` and passing this only in `separatevars` we are not able to separate it so I think we can pass simplify(factor) every time in `separatevars `so that we can get a simplified equation and then we can easily separate the equation.





### Other comments


#### Release Notes

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<!-- BEGIN RELEASE NOTES -->
- simplify
  - remove a bug from separatevars and make it more robust
<!-- END RELEASE NOTES -->

[oscargus]

It would be nice to see the impact on simplify here. Is there a test one can add that works now and not earlier that illustrates the benefit?

[RituRajSingh878]

Is there a test one can add that works now and not earlier that illustrates the benefit?

the newly added test case was not working, now they are working

[RituRajSingh878]

although we can separate the equation - eq=exp(x**2+y**2)*(cos(x-y)+cos(x+y) now but not able to separate this((cos(x-y)+cos(x+y)) equation. So I think We nedd to do some extra work. maybe we can try to separate the equation just by passing and if it not separable then we can try again by passing simplify(eq)

[oscarbenjamin]

If this is going to call something like simplify then there probably needs to be an option to prevent that. At the moment separatevars is a fairly simple low-level routine which is used in a few places (not many) and we should be careful that this doesn't slow those cases down.

[oscarbenjamin]

It's probably better to rewrite sin/cos as exp:

In [28]: e = cos(x+y)+cos(x-y)                                                                                                    

In [29]: factor(e.rewrite(exp)).rewrite(sin)                                                                                      
Out[29]: 2⋅cos(x)⋅cos(y)

[RituRajSingh878]

It's probably better to rewrite sin/cos as exp:

In [28]: e = cos(x+y)+cos(x-y)                                                                                                    

In [29]: factor(e.rewrite(exp)).rewrite(sin)                                                                                      
Out[29]: 2⋅cos(x)⋅cos(y)

It may help but sometime it will make the equation very complex like-

In [28]: e=sin(x+y)/cos(x+y)

In [29]: factor(e.rewrite(exp)).rewrite(sin)
Out[29]: 
ⅈ⋅((-ⅈ⋅sin(x) + cos(x))⋅(-ⅈ⋅sin(y) + cos(y)) - (ⅈ⋅sin(x) + cos(x))⋅(ⅈ⋅sin(y) +
──────────────────────────────────────────────────────────────────────────────
  (-ⅈ⋅sin(x) + cos(x))⋅(-ⅈ⋅sin(y) + cos(y)) + (ⅈ⋅sin(x) + cos(x))⋅(ⅈ⋅sin(y) + 

 cos(y)))
─────────
cos(y))  

in this case, we have to deal with the more complex equation than given one so I think we can't do every time.

[oscarbenjamin]

In [28]: e=sin(x+y)/cos(x+y)

Is this example separable?

[RituRajSingh878]

In [28]: e=sin(x+y)/cos(x+y)

Is this example separable?

No
I am just taking an example to show that a simpler example can be a complex due to this.

[czgdp1807]

Any news? Can it be taken over by somebody else?

[smichr]

I would recommend against making separatevars be too smart. Some routines should just do a straigtforward task fast and well. It is the user's (or high-level routine's) job to try rewriting to make it work.