54.3.253 problem 1269

Internal problem ID [12548]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1269
Date solved : Friday, October 03, 2025 at 03:38:14 AM
CAS classification : [_Jacobi]

\begin{align*} 2 x \left (x -1\right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 78

ode:=2*x*(x-1)*diff(diff(y(x),x),x)+((2*v+5)*x-2*v-3)*diff(y(x),x)+(v+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (c_1 \Gamma \left (v +\frac {3}{2}\right )^{2} \operatorname {LegendreP}\left (-\frac {1}{2}, -\frac {1}{2}-v , \frac {-x -1}{x -1}\right )+\pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) c_2 \sec \left (\pi v \right ) \left (v +\frac {1}{2}\right )\right ) x^{-\frac {1}{4}-\frac {v}{2}}}{\sqrt {1-x}\, \Gamma \left (v +\frac {3}{2}\right )} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 59

ode=(1 + v)*y[x] + (-3 - 2*v + (5 + 2*v)*x)*D[y[x],x] + 2*(-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 {Hypergeometric2F1}\left (\frac {1}{2},v+1,v+\frac {3}{2},x\right )-i c_2 i^{-2 v} x^{-v-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-v,\frac {1}{2}-v,x\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
ode = Eq(2*x*(x - 1)*Derivative(y(x), (x, 2)) + (v + 1)*y(x) + (-2*v + x*(2*v + 5) - 3)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False