55.33.25 problem 234

Internal problem ID [14008]
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 : 234
Date solved : Thursday, October 02, 2025 at 09:08:48 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 174

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

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

✓ Mathematica. Time used: 0.782 (sec). Leaf size: 199

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

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

✗ Sympy

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

False