55.33.23 problem 232

Internal problem ID [14006]
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 : 232
Date solved : Thursday, October 02, 2025 at 09:08:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 104

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

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

✓ Mathematica. Time used: 19.535 (sec). Leaf size: 141

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

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

✗ Sympy

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

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