23.3.364 problem 368

Internal problem ID [6078]
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 : 368
Date solved : Friday, October 03, 2025 at 01:46:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 26

ode=-(p*(1 + p)*y[x]) + 2*x*D[y[x],x] + (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}(p,i x)+c_2 \operatorname {LegendreQ}(p,i x) \end{align*}

✗ Sympy

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

False