54.3.240 problem 1256

Internal problem ID [12535]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1256
Date solved : Friday, October 03, 2025 at 03:30:48 AM
CAS classification : [_Jacobi]

\begin{align*} x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 51

ode:=x*(x-1)*diff(diff(y(x),x),x)+(2*x-1)*diff(y(x),x)-v*(v+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to c_1 \operatorname {LegendreP}(v,2 x-1)+c_2 \operatorname {LegendreQ}(v,2 x-1) \end{align*}

✗ Sympy

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

False