23.3.571 problem 579

Internal problem ID [6285]
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 : 579
Date solved : Friday, October 03, 2025 at 01:58:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (a \left (1+a \right ) \left (1-x \right )+b^{2} x \right ) y+2 \left (1-3 x \right ) \left (1-x \right ) x y^{\prime }+4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 93

ode:=-(a*(a+1)*(1-x)+b^2*x)*y(x)+2*(1-3*x)*(1-x)*x*diff(y(x),x)+4*x^2*(-x^2+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x +1\right )^{3} \left (c_1 \,x^{-\frac {a}{2}} \operatorname {HeunG}\left (-1, \frac {1}{4} a^{2}-\frac {1}{4} b^{2}-\frac {7}{4} a +\frac {3}{2}, 3-\frac {a}{2}, \frac {1}{2}-\frac {a}{2}, -a +\frac {1}{2}, 0, x\right )+c_2 \,x^{\frac {1}{2}+\frac {a}{2}} \operatorname {HeunG}\left (-1, \frac {1}{4} a^{2}-\frac {1}{4} b^{2}+\frac {9}{4} a +\frac {7}{2}, 1+\frac {a}{2}, \frac {7}{2}+\frac {a}{2}, a +\frac {3}{2}, 0, x\right )\right ) \]

✗ Mathematica

ode=-((a*(1 + a)*(1 - x) + b^2*x)*y[x]) + 2*(1 - 3*x)*(1 - x)*x*D[y[x],x] + 4*x^2*(1 - x^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(4*x**2*(1 - x**2)*Derivative(y(x), (x, 2)) + x*(1 - x)*(2 - 6*x)*Derivative(y(x), x) + (-a*(1 - x)*(a + 1) - b**2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None