Solver for Integrable series of special Riccati equation.

[yuri-karadzhov]

The solver for special Riccati equation integrable series will be added to sympy.

Original issue for #5348: http://code.google.com/p/sympy/issues/detail?id=2249 Original author: https://code.google.com/u/110328265044803872166/

[asmeurer]

**Status:** Duplicate  
**Mergedinto:** 2250  

Original comment: http://code.google.com/p/sympy/issues/detail?id=2249#c1 Original author: https://code.google.com/u/asmeurer@gmail.com/

[asmeurer]

Apparently this is actually separate from issue #5349 .

Status: Accepted
Mergedinto:

Referenced issues: #5349 Original comment: http://code.google.com/p/sympy/issues/detail?id=2249#c2 Original author: https://code.google.com/u/asmeurer@gmail.com/

[yuri-karadzhov]

After the changes it will be possible to solve special Riccati euation

a*y'+b*y**2+c*x**alpha=0

where a, b, c != 0 and there's two infinite series for alpha when the equation is integrable in elementary functions.

Original comment: http://code.google.com/p/sympy/issues/detail?id=2249#c3 Original author: https://code.google.com/u/110328265044803872166/

[asmeurer]

**Status:** Valid  

Original comment: http://code.google.com/p/sympy/issues/detail?id=2249#c4 Original author: https://code.google.com/u/asmeurer@gmail.com/

[oscarbenjamin]

Currently this equation is matched by the 1st_power_series solver but it just returns oo. The lie_group solver also matches but then fails with NotImplementedError:

In [21]: a, b, c, alpha = symbols('a, b, c, alpha')                                                                                            

In [22]: x = Symbol('x')                                                                                                                       

In [23]: y = Function('y')                                                                                                                     

In [24]: eq = a*y(x).diff(x) + b*y(x)**2 + c*x**alpha                                                                                          

In [25]: eq                                                                                                                                    
Out[25]: 
  d             2         α
a⋅──(y(x)) + b⋅y (x) + c⋅x 
  dx                       

In [26]: dsolve(eq)                                                                                                                            
Out[26]: y(x) = ∞

In [27]: classify_ode(eq, y(x))                                                                                                                
Out[27]: ('1st_power_series', 'lie_group')

In [28]: dsolve(eq, hint='lie_group')                                                                                                          
---------------------------------------------------------------------------
NotImplementedError 

It looks like Ricatti equations can always be reduced to linear ODEs by substitution: https://en.wikipedia.org/wiki/Riccati_equation I think it would be useful if the ODE module actually implemented substitutions explicitly since it would make it very easy to implement solvers for equations like this.

[naveensaigit]

I think the substitution would be helpful if we implement the Kovacic algorithm. That way, any Riccati equation could be transformed to a linear second order equation which can then be solved by Kovacic. Another thing we could also use is the fact that knowing a particular solution gives the second solution very easily. But this would be a lot of special case code.

[oscarbenjamin]

We should definitely implement the Kovacic algorithm and make sure all cases of the Riccati equation are covered.

Right now the most immediately useful thing would be to fix #17590.