23.3.499 problem 505

Internal problem ID [6213]
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 : 505
Date solved : Tuesday, September 30, 2025 at 02:36:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-1+a \right ) \left (a +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: 0.016 (sec). Leaf size: 39

ode:=(a-1)*(a+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);
 

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

✓ Mathematica. Time used: 0.415 (sec). Leaf size: 53

ode=(-1 + a)*(a + 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 {c_2 \operatorname {Hypergeometric2F1}\left (\frac {a-1}{2},\frac {1}{2} (-a-b),\frac {a+1}{2},x^2\right )}{a-1}+c_1 x^{1-a} \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*(a - 1)*(a + 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