55.33.8 problem 218

Internal problem ID [13991]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 218
Date solved : Friday, October 03, 2025 at 07:23:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 260

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

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

✓ Mathematica. Time used: 165.204 (sec). Leaf size: 413606

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

NotImplementedError : The given ODE a*x**2*(x - 1)**2*Derivative(y(x), (x, 2)) + (b*x**2 + c*x + d)*y(x) cannot be solved by the hypergeometric method