added hypergeometic method for 1F1 type ode

[RituRajSingh878]

References to other Issues or PRs

related #17603

Brief description of what is fixed or changed

By implementing this method, any differential equation that can be transform into eq = x*f(x).diff(x, 2) + (c-x)*f(x).diff(x) -a*f(x) by applying the following transformations

1- x --> x**k
2- x --> (a*x + b)/(c*x+d)
3- y --> P(x)*y

then the given equation can be solve.

It will cover a class of ode-

In [5]: eq = x*f(x).diff(x, 2) + (c-x)*f(x).diff(x) -a*f(x)

In [6]: eq
Out[6]: 
              2                         
             d                  d       
-a⋅f(x) + x⋅───(f(x)) + (c - x)⋅──(f(x))
              2                 dx      
            dx                          

Which is solvable in the series solution.

Other comments

Work is still in progress.
sources-
https://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation
https://drive.google.com/file/d/1wcJlEd6DMTRHAprH2N2oXaCHpa8PLDwp/view

Release Notes

• solvers
  • added method in dsolve to solve 2nd order ode in 1F1 hyper function

[sympy-bot]

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• solvers
  • added method in dsolve to solve 2nd order ode in 1F1 hyper function (#17750 by @RituRajSingh878)

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#### References to other Issues or PRs
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related #17603 
#### Brief description of what is fixed or changed
By implementing this method, any differential equation that can be transform into `eq = x*f(x).diff(x, 2) + (c-x)*f(x).diff(x) -a*f(x)` by applying the following transformations 
```
1- x --> x**k
2- x --> (a*x + b)/(c*x+d)
3- y --> P(x)*y
```
then the given equation can be solve.

It will cover a class of ode- 
```
In [5]: eq = x*f(x).diff(x, 2) + (c-x)*f(x).diff(x) -a*f(x)

In [6]: eq
Out[6]: 
              2                         
             d                  d       
-a⋅f(x) + x⋅───(f(x)) + (c - x)⋅──(f(x))
              2                 dx      
            dx                          


```
Which is solvable in the series solution.



#### Other comments
Work is  still in progress.
sources-
https://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation
https://drive.google.com/file/d/1wcJlEd6DMTRHAprH2N2oXaCHpa8PLDwp/view
#### Release Notes

<!-- Write the release notes for this release below. See
https://github.com/sympy/sympy/wiki/Writing-Release-Notes for more information
on how to write release notes. The bot will check your release notes
automatically to see if they are formatted correctly. -->

<!-- BEGIN RELEASE NOTES -->
- solvers
  - added method in dsolve to solve 2nd order ode in 1F1 hyper function
<!-- END RELEASE NOTES -->

[codecov]

Codecov Report

Merging #17750 into master will increase coverage by 9.538%.
The diff coverage is 0%.

@@              Coverage Diff              @@
##            master    #17750       +/-   ##
=============================================
+ Coverage   65.215%   74.753%   +9.538%     
=============================================
  Files          636       636               
  Lines       166193    166203       +10     
  Branches     39068     39073        +5     
=============================================
+ Hits        108383    124243    +15860     
+ Misses       51856     36449    -15407     
+ Partials      5954      5511      -443

[RituRajSingh878]

I am trying to implement the solution and I am using whitm but I am getting the following error whenever I am using a variable inside the whitm.

In [14]: from mpmath import *

In [15]: whitm(1/2, 2/3, x**2/3 + 1)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-15-1ab057ac9216> in <module>()
----> 1 whitm(1/2, 2/3, x**2/3 + 1)

/home/rituraj/.local/lib/python3.6/site-packages/mpmath/ctx_mp_python.py in f_wrapped(ctx, *args, **kwargs)
   1029             def f_wrapped(ctx, *args, **kwargs):
   1030                 convert = ctx.convert
-> 1031                 args = [convert(a) for a in args]
   1032                 prec = ctx.prec
   1033                 try:

/home/rituraj/.local/lib/python3.6/site-packages/mpmath/ctx_mp_python.py in <listcomp>(.0)
   1029             def f_wrapped(ctx, *args, **kwargs):
   1030                 convert = ctx.convert
-> 1031                 args = [convert(a) for a in args]
   1032                 prec = ctx.prec
   1033                 try:

/home/rituraj/.local/lib/python3.6/site-packages/mpmath/ctx_mp_python.py in convert(ctx, x, strings)
    667             try: return ctx.make_mpf(from_Decimal(x, prec, rounding))
    668             except: pass
--> 669         return ctx._convert_fallback(x, strings)
    670 
    671     def npconvert(ctx, x):

/home/rituraj/.local/lib/python3.6/site-packages/mpmath/ctx_mp.py in _convert_fallback(ctx, x, strings)
    632             else:
    633                 raise ValueError("can only create mpf from zero-width interval")
--> 634         raise TypeError("cannot create mpf from " + repr(x))
    635 
    636     def mpmathify(ctx, *args, **kwargs):

TypeError: cannot create mpf from x**2/3 + 1

[oscarbenjamin]

You are importing whitm from mpmath which is a numerical library. You can't use symbols in numerical code.

[RituRajSingh878]

Is there any way I can use whitm with a variable?

[oscarbenjamin]

Not if whitm is an mpmath function. If you need this function then you can create a SymPy class to represent the function if there isn't one already. I guess it should go in sympy.functions.special.hyper.

[czgdp1807]

@RituRajSingh878 @oscarbenjamin Is this related to this year's GSoC ODE project by @mijo2 ?