23.3.544 problem 550

Internal problem ID [6258]
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 : 550
Date solved : Friday, October 03, 2025 at 01:57:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \left (1+a \right ) y-2 x^{3} y^{\prime }+x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 153

ode:=a*(a+1)*y(x)-2*x^3*diff(y(x),x)+x^2*(-x^2+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left (c_1 \,x^{-\frac {\sqrt {-4 a^{2}-4 a +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {-4 a^{2}-4 a +1}}{4}, \frac {1}{4}-\frac {\sqrt {-4 a^{2}-4 a +1}}{4}\right ], \left [1-\frac {\sqrt {-4 a^{2}-4 a +1}}{2}\right ], x^{2}\right )+c_2 \,x^{\frac {\sqrt {-4 a^{2}-4 a +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {\sqrt {-4 a^{2}-4 a +1}}{4}, \frac {1}{4}+\frac {\sqrt {-4 a^{2}-4 a +1}}{4}\right ], \left [1+\frac {\sqrt {-4 a^{2}-4 a +1}}{2}\right ], x^{2}\right )\right ) \]

✓ Mathematica. Time used: 0.135 (sec). Leaf size: 227

ode=a*(1 + a)*y[x] - 2*x^3*D[y[x],x] + x^2*(1 - x^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (-1)^{\frac {1}{4} \left (1-\sqrt {-4 a^2-4 a+1}\right )} x^{\frac {1}{2}-\frac {1}{2} \sqrt {-4 a^2-4 a+1}} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4}-\frac {1}{4} \sqrt {-4 a^2-4 a+1},\frac {3}{4}-\frac {1}{4} \sqrt {-4 a^2-4 a+1},1-\frac {1}{2} \sqrt {-4 a^2-4 a+1},x^2\right )+i^{\sqrt {-4 a^2-4 a+1}} c_2 x^{\sqrt {-4 a^2-4 a+1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (\sqrt {-4 a^2-4 a+1}+1\right ),\frac {1}{4} \left (\sqrt {-4 a^2-4 a+1}+3\right ),\frac {1}{2} \left (\sqrt {-4 a^2-4 a+1}+2\right ),x^2\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*(a + 1)*y(x) - 2*x**3*Derivative(y(x), x) + x**2*(1 - x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None