54.3.358 problem 1375

Internal problem ID [12653]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1375
Date solved : Friday, October 03, 2025 at 03:45:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 34

ode=D[y[x],{x,2}] == -(((4*a*(a - n)*x^2 - (2*a + (-n + v)*(1 + n + v))*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*(1 - 2*a + n)*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-\frac {n}{2}} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \end{align*}

✗ Sympy

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

False