54.3.317 problem 1334

Internal problem ID [12612]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1334
Date solved : Friday, October 03, 2025 at 03:43:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (\left (a +1\right ) x -1\right ) y^{\prime }}{x \left (x -1\right )}-\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )} \end{align*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 89

ode:=diff(diff(y(x),x),x) = -((a+1)*x-1)/x/(x-1)*diff(y(x),x)-1/4*((a^2-b^2)*x+c^2)/x^2/(x-1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x^{-\frac {c}{2}} \operatorname {hypergeom}\left (\left [-\frac {a}{2}-\frac {b}{2}-\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}-\frac {c}{2}+1\right ], \left [1-c \right ], x\right ) c_2 +x^{\frac {c}{2}} \operatorname {hypergeom}\left (\left [-\frac {a}{2}-\frac {b}{2}+\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}+\frac {c}{2}+1\right ], \left [1+c \right ], x\right ) c_1 \right ) \left (x -1\right )^{1-a} \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 89

ode=D[y[x],{x,2}] == -1/4*((c^2 + (a^2 - b^2)*x)*y[x])/((-1 + x)*x^2) - ((-1 + (1 + a)*x)*D[y[x],x])/((-1 + x)*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to i^{-c} x^{-c/2} \left (i^{2 c} c_2 x^c \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (a-b+c),\frac {1}{2} (a+b+c),c+1,x\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (a-b-c),\frac {1}{2} (a+b-c),1-c,x\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (x*(a + 1) - 1)*Derivative(y(x), x)/(x*(x - 1)) + (c**2 + x*(a**2 - b**2))*y(x)/(4*x**2*(x - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None