31.20 problem 201

Internal problem ID [11024]
Internal file name [OUTPUT/10281_Sunday_December_31_2023_03_57_10_PM_26080740/index.tex]

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 Equation of form \((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 201.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

\[ \boxed {\left (a \,x^{3}+x^{2}+b \right ) y^{\prime \prime }+a^{2} x \left (x^{2}-b \right ) y^{\prime }-a^{3} b x y=0} \]

Maple trace Kovacic algorithm successful

`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 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
   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 @ Moebius 
      -> 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`
 

✓ Solution by Maple

Time used: 0.562 (sec). Leaf size: 79

dsolve((a*x^3+x^2+b)*diff(y(x),x$2)+a^2*x*(x^2-b)*diff(y(x),x)-a^3*b*x*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-a x} \left (c_{2} \left (\int {\mathrm e}^{a \left (\int \frac {a^{2} x^{4}+2 a \,x^{3}+\left (a^{2} b +2\right ) x^{2}+4 a b x +2 b}{\left (a \,x^{3}+x^{2}+b \right ) \left (a x +2\right )}d x \right )}d x \right )+c_{1} \right ) \left (a x +2\right ) \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[(a*x^3+x^2+b)*y''[x]+a^2*x*(x^2-b)*y'[x]-a^3*b*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Timed out