23.3.560 problem 567

Internal problem ID [6274]
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 : 567
Date solved : Friday, October 03, 2025 at 01:58:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 227

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

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

✓ Mathematica. Time used: 67.427 (sec). Leaf size: 215249

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

Too large to display

✗ Sympy

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

NotImplementedError : The given ODE x**2*(1 - x)**2*Derivative(y(x), (x, 2)) + (a + b*x + c*x**2)*y(