55.33.19 problem 228

Internal problem ID [14002]
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 : 228
Date solved : Thursday, October 02, 2025 at 09:08:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 148

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

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

✓ Mathematica. Time used: 2.051 (sec). Leaf size: 91

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

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

✗ Sympy

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

False