54.3.312 problem 1329

Internal problem ID [12607]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1329
Date solved : Friday, October 03, 2025 at 03:43:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.186 (sec). Leaf size: 64

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

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

✓ Mathematica. Time used: 1.038 (sec). Leaf size: 67

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

\begin{align*} y(x)&\to c_2 x^{1-\gamma } \text {HeunG}[a,q-(\gamma -1) ((a-1) \delta +\alpha +\beta -\gamma +1),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\delta ,x]+c_1 \text {HeunG}[a,q,\alpha ,\beta ,\gamma ,\delta ,x] \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None