2.2.46 Problem 45

Internal problem ID [8850]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 45
Date solved : Sunday, March 30, 2025 at 01:42:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

✓ Maple. Time used: 0.044 (sec). Leaf size: 57

ode:=diff(diff(y(x),x),x)-x^2*diff(y(x),x)-x^2*y(x)-x^3-x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y={\mathrm{e}}^{-x}\mathrm{HeunT}(3^{2/3},3,23^{1/3},\frac{3^{2/3}x}{3})c_2+\mathrm{HeunT}(3^{2/3},-3,23^{1/3},-\frac{3^{2/3}x}{3}){\mathrm{e}}^{\frac{x(x^2+3)}{3}}{c}_1-x$$

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   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 
   <- No Liouvillian solutions exists 
   -> 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 
   trying a solution in terms of MeijerG functions 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ 
er @ Moebius 
   <- Heun successful: received ODE is equivalent to the  HeunT  ODE, case  c =\ 
 0 
<- solving first the homogeneous part of the ODE successful
 

✗ Mathematica

ode=D[y[x],{x,2}]-x^2*D[y[x],x]-x^2*y[x]-x^3-x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 - x**2*y(x) - x**2*Derivative(y(x), x) - x**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x + y(x) + Derivative(y(x), x) + 1 - Derivative(y(x), (x, 2))/x**2 cannot be solved by the factorable group method