54.3.372 problem 1389

Internal problem ID [12667]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1389
Date solved : Friday, October 03, 2025 at 03:45:42 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 {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 87

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-1)^2-4*n^2*x)/x^2/(x-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.591 (sec). Leaf size: 109

ode=D[y[x],{x,2}] == -1/4*((-(v*(1 + v)*(-1 + x)^2) - 4*n^2*x)*y[x])/((-1 + x)^2*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 (-1)^{-v} (x-1)^{n+\frac {1}{2}} x^{\frac {1}{4}-\frac {v}{2}} e^{-\frac {1}{4} \int \frac {1-3 x}{x-x^2} \, dx} \left (c_1 (-1)^v x^{v+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (n+\frac {1}{2},n+v+1,v+\frac {3}{2},x\right )-i c_2 \operatorname {Hypergeometric2F1}\left (n+\frac {1}{2},n-v,\frac {1}{2}-v,x\right )\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None