2.32.25 Problem 207

Internal problem ID [13867]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 207
Date solved : Friday, December 19, 2025 at 05:45:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 x a +2 b +\lambda \right ) y&=0 \\ \end{align*}

2.32.25.1 ✓ Maple. Time used: 0.050 (sec). Leaf size: 99

ode:=2*(a*x^3+b*x^2+c*x+d)*diff(diff(y(x),x),x)+3*(3*a*x^2+2*b*x+c)*diff(y(x),x)+(6*a*x+2*b+lambda)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (c_1 \,{\mathrm e}^{\sqrt {2}\, \sqrt {-\frac {\lambda }{a}}\, \int \frac {1}{\sqrt {\frac {a \,x^{3}+b \,x^{2}+c x +d}{a}}}d x}+c_2 \right ) {\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\frac {\lambda }{a}}\, \int \frac {1}{\sqrt {\frac {a \,x^{3}+b \,x^{2}+c x +d}{a}}}d x}{2}}}{\sqrt {a \,x^{3}+b \,x^{2}+c x +d}} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
   A Liouvillian solution exists 
   Group is reducible or imprimitive 
   Solution has integrals. Trying a special function solution free of integrals\ 
... 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Mo\ 
ebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
   No special function solution was found. 
<- Kovacics algorithm successful
 

2.32.25.2 ✗ Mathematica

ode=2*(a*x^3+b*x^2+c*x+d)*D[y[x],{x,2}]+3*(3*a*x^2+2*b*x+c)*D[y[x],x]+(6*a*x+2*b+\[Lambda])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

2.32.25.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq((6*a*x + 2*b + lambda_)*y(x) + (9*a*x**2 + 6*b*x + 3*c)*Derivative(y(x), x) + (2*a*x**3 + 2*b*x**2 + 2*c*x + 2*d)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_ordinary')