54.3.316 problem 1333

Internal problem ID [12611]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1333
Date solved : Friday, October 03, 2025 at 03:43:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {v \left (v +1\right ) y}{4 x^{2}} \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 74

ode:=diff(diff(y(x),x),x) = -1/2/x*(3*x-1)/(x-1)*diff(y(x),x)+1/4*v*(v+1)/x^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 70

ode=D[y[x],{x,2}] == (v*(1 + v)*y[x])/(4*x^2) - ((-1 + 3*x)*D[y[x],x])/(2*(-1 + x)*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 i^{-v} x^{-v/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-v,\frac {1}{2}-v,x\right )+c_2 i^{v+1} x^{\frac {v+1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},v+1,v+\frac {3}{2},x\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
ode = Eq(-v*(v + 1)*y(x)/(4*x**2) + Derivative(y(x), (x, 2)) + (3*x - 1)*Derivative(y(x), x)/(2*x*(x - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None