23.3.463 problem 468

Internal problem ID [6177]
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 : 468
Date solved : Friday, October 03, 2025 at 01:48:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.122 (sec). Leaf size: 58

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

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

✗ Mathematica

ode=(a + b*x + c*x^2)*y[x] + 2*(1 - 2*x)*D[y[x],x] + 4*(1 - x)*x*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") 
c = symbols("c") 
y = Function("y") 
ode = Eq(x*(4 - 4*x)*Derivative(y(x), (x, 2)) + (2 - 4*x)*Derivative(y(x), x) + (a + b*x + c*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False