23.3.550 problem 556

Internal problem ID [6264]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 556
Date solved : Friday, October 03, 2025 at 01:58:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 37

ode:=-(a^2-k*(-x^2+1))*y(x)-2*x*(-x^2+1)*diff(y(x),x)+(-x^2+1)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {LegendreP}\left (\frac {\sqrt {1+4 k}}{2}-\frac {1}{2}, a , x\right )+c_2 \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 k}}{2}-\frac {1}{2}, a , x\right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 48

ode=-((a^2 - k*(1 - x^2))*y[x]) - 2*x*(1 - x^2)*D[y[x],x] + (1 - x^2)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 P_{\frac {1}{2} \left (\sqrt {4 k+1}-1\right )}^a(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 k+1}-1\right )}^a(x) \end{align*}

✗ Sympy

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

False