55.33.6 problem 216

Internal problem ID [13989]
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 : 216
Date solved : Thursday, October 02, 2025 at 09:08:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 67

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

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

✓ Mathematica. Time used: 17.986 (sec). Leaf size: 121

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

\begin{align*} y(x)&\to \frac {x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} \left (a c_1 \sqrt {1-\frac {4 b}{a^2}} x^{\sqrt {1-\frac {4 b}{a^2}}}+c_2 (x-a)^{\sqrt {1-\frac {4 b}{a^2}}}\right )}{a \sqrt {1-\frac {4 b}{a^2}}} \end{align*}

✗ Sympy

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

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