23.3.458 problem 463

Internal problem ID [6172]
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 : 463
Date solved : Friday, October 03, 2025 at 01:48:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (4 p^{2}+1\right ) y-8 x y^{\prime }+4 \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 25

ode:=-(4*p^2+1)*y(x)-8*x*diff(y(x),x)+4*(-x^2+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {LegendreP}\left (i p -\frac {1}{2}, x\right )+c_2 \operatorname {LegendreQ}\left (i p -\frac {1}{2}, x\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 34

ode=-((1 + 4*p^2)*y[x]) - 8*x*D[y[x],x] + 4*(1 - x^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
p = symbols("p") 
y = Function("y") 
ode = Eq(-8*x*Derivative(y(x), x) + (4 - 4*x**2)*Derivative(y(x), (x, 2)) + (-4*p**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False