61.31.23 problem 204

Internal problem ID [12625]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 204
Date solved : Monday, March 31, 2025 at 06:09:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x -1\right ) \left (x -a \right ) y^{\prime \prime }+\left (\left (\alpha +\beta +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +d \right )-a \right ) x +a \gamma \right ) y^{\prime }+\left (\alpha \beta x -q \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.199 (sec). Leaf size: 82

ode:=x*(x-1)*(x-a)*diff(diff(y(x),x),x)+((alpha+beta+1)*x^2-(alpha+beta+1+a*(gamma+d)-a)*x+a*gamma)*diff(y(x),x)+(alpha*beta*x-q)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.321 (sec). Leaf size: 85

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

\[ y(x)\to c_2 x^{1-\gamma } \text {HeunG}\left [a,q-(\gamma -1) (a (d-1)+\alpha +\beta -\gamma +1),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\frac {a (d-1)}{a-1},x\right ]+c_1 \text {HeunG}\left [a,q,\alpha ,\beta ,\gamma ,\frac {a (d-1)}{a-1},x\right ] \]

✗ Sympy

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

ValueError : Expected Expr or iterable but got None