54.3.357 problem 1374

Internal problem ID [12652]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1374
Date solved : Friday, October 03, 2025 at 03:45:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 26

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

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

✗ Sympy

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

False