23.3.525 problem 531

Internal problem ID [6239]
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 : 531
Date solved : Friday, October 03, 2025 at 01:57:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 46

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

\[ y = \left (-1+x \right ) \left (c_1 \operatorname {HeunC}\left (-\frac {3}{2}, -\frac {1}{2}, 1, \frac {3}{8}-\frac {b}{4}, -\frac {a}{4}+\frac {1}{8}, x\right )+c_2 \operatorname {HeunC}\left (-\frac {3}{2}, \frac {1}{2}, 1, \frac {3}{8}-\frac {b}{4}, -\frac {a}{4}+\frac {1}{8}, x\right ) \sqrt {x}\right ) \]

✓ Mathematica. Time used: 0.192 (sec). Leaf size: 61

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

\begin{align*} y(x)&\to c_2 \sqrt {x} \text {HeunC}\left [\frac {a-3}{4},\frac {1}{4} (-b-3),\frac {3}{2},0,-\frac {3}{2},x\right ]+c_1 \text {HeunC}\left [\frac {a}{4},-\frac {b}{4},\frac {1}{2},0,-\frac {3}{2},x\right ] \end{align*}

✗ Sympy

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

False