2.2.45 Problem 44

Internal problem ID [8849]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 44
Date solved : Sunday, March 30, 2025 at 01:42:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

✓ Maple. Time used: 0.005 (sec). Leaf size: 44

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

$$y={\mathrm{e}}^{\frac{x^3}{6}}\sqrt{x}\mathrm{BesselI}(\frac{1}{6},\frac{x^3}{6})c_2+{\mathrm{e}}^{\frac{x^3}{6}}\sqrt{x}\mathrm{BesselK}(\frac{1}{6},\frac{x^3}{6})c_1-\frac{x}{2}$$

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 
      <- Bessel successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful
 

✓ Mathematica. Time used: 0.318 (sec). Leaf size: 224

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

$$y(x)\to -\frac{e^{\frac{x^3}{6}}(12{\left(x^3\right)}^{5/6}\mathrm{Gamma}\left(\frac{1}{6}\right)\mathrm{Gamma}\left(\frac{7}{6}\right)\mathrm{BesselI}(\frac{1}{6},\frac{x^3}{6}){}_1{F}_1(-\frac{2}{3};-\frac{1}{3};-\frac{x^3}{3})+\sqrt[3]{2}{3}^{2/3}\sqrt[6]{x^3}{x}^6\mathrm{Gamma}\left(\frac{1}{6}\right)\mathrm{Gamma}\left(\frac{5}{6}\right)\mathrm{BesselI}(-\frac{1}{6},\frac{x^3}{6}){}_1{F}_1(\frac{2}{3};\frac{7}{3};-\frac{x^3}{3})-4\mathrm{Gamma}\left(\frac{7}{6}\right)(6\sqrt[3]{2}{3}^{2/3}{c}_1{x}^{5/2}\mathrm{Gamma}\left(\frac{5}{6}\right)\mathrm{BesselI}(-\frac{1}{6},\frac{x^3}{6})+\mathrm{Gamma}\left(\frac{1}{6}\right)(3{\left(x^3\right)}^{5/6}+2\sqrt[3]{-1}{3}^{2/3}{c}_2{x}^{5/2})\mathrm{BesselI}(\frac{1}{6},\frac{x^3}{6})))}{242^{2/3}{3}^{5/6}{x}^2\mathrm{Gamma}\left(\frac{7}{6}\right)}$$

✗ Sympy

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

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