61.32.25 problem 234

Internal problem ID [12656]
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 : Monday, March 31, 2025 at 06:49:38 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: 0.421 (sec). Leaf size: 154

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]
 

\[ y(x)\to \exp \left (\int _1^x\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right ) \]

✗ 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