54.3.308 problem 1325

Internal problem ID [12603]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1325
Date solved : Friday, October 03, 2025 at 03:41:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 86

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

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

✓ Mathematica. Time used: 0.191 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to (-1)^{\beta } c_2 x^{\beta } \operatorname {Hypergeometric2F1}(a+\beta ,b+\beta ,-\alpha +\beta +1,x)+(-1)^{\alpha } c_1 x^{\alpha } \operatorname {Hypergeometric2F1}(a+\alpha ,b+\alpha ,\alpha -\beta +1,x) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None