61.30.23 problem 171

Internal problem ID [12592]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 171
Date solved : Monday, March 31, 2025 at 05:52:37 AM
CAS classification : [_Jacobi]

\begin{align*} x \left (x -1\right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x -\gamma \right ) y^{\prime }+\alpha \beta y&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 44

ode:=x*(x-1)*diff(diff(y(x),x),x)+((alpha+beta+1)*x-gamma)*diff(y(x),x)+alpha*beta*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [\alpha , \beta \right ], \left [\gamma \right ], x\right )+c_2 \,x^{1-\gamma } \operatorname {hypergeom}\left (\left [\beta +1-\gamma , \alpha +1-\gamma \right ], \left [2-\gamma \right ], x\right ) \]

✓ Mathematica. Time used: 0.184 (sec). Leaf size: 49

ode=x*(x-1)*D[y[x],{x,2}]+((\[Alpha]+\[Beta]+1)*x-\[Gamma])*D[y[x],x]+\[Alpha]*\[Beta]*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}(\alpha ,\beta ,\gamma ,x)-(-1)^{-\gamma } c_2 x^{1-\gamma } \operatorname {Hypergeometric2F1}(\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,x) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
Gamma = symbols("Gamma") 
y = Function("y") 
ode = Eq(Alpha*BETA*y(x) + x*(x - 1)*Derivative(y(x), (x, 2)) + (-Gamma + x*(Alpha + BETA + 1))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None