23.3.500 problem 506

Internal problem ID [6214]
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 : 506
Date solved : Friday, October 03, 2025 at 01:56:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.051 (sec). Leaf size: 50

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

\[ y = c_1 \left (x^{2} \left (-1+b \right )+1+a \right )+c_2 \,x^{1-a} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {a}{2}, 1-\frac {b}{2}-\frac {a}{2}\right ], \left [\frac {3}{2}-\frac {a}{2}\right ], x^{2}\right ) \]

✓ Mathematica. Time used: 0.9 (sec). Leaf size: 106

ode=2*(1 - b)*x*y[x] + (a + b*x^2)*D[y[x],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 \frac {x^{-a} \left (a+(b-1) x^2+1\right ) \left ((a-1) (a+1)^2 c_1 x^a-c_2 x \operatorname {AppellF1}\left (\frac {1-a}{2},\frac {1}{2} (-a-b),2,\frac {3-a}{2},x^2,-\frac {(b-1) x^2}{a+1}\right )\right )}{(a-1) (a+1)^2 (a+4 b-3)} \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None