2.2.48 Problem 47

Internal problem ID [8851]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 47
Date solved : Friday, April 25, 2025 at 05:14:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)-1/x*diff(y(x),x)-x*y(x)-x^2-1/x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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: 0.449 (sec). Leaf size: 253

ode=D[y[x],{x,2}]-1/x*D[y[x],x]-x*y[x]-x^2-1/x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to \frac{\frac{3\sqrt[6]{3}\pi \mathrm{Gamma}(-\frac{1}{3})(3\mathrm{AiryAiPrime}(x)+\sqrt{3}\mathrm{AiryBiPrime}(x)){}_1{F}_2(-\frac{1}{3};\frac{1}{3},\frac{2}{3};\frac{x^3}{9})}{x\mathrm{Gamma}\left(\frac{2}{3}\right)}+\frac{\frac{\sqrt[3]{3}\pi x\mathrm{Gamma}{\left(\frac{1}{3}\right)}^2(\sqrt{3}\mathrm{AiryAiPrime}(x)-\mathrm{AiryBiPrime}(x)){}_1{F}_2(\frac{1}{3};\frac{4}{3},\frac{5}{3};\frac{x^3}{9})}{\mathrm{Gamma}\left(\frac{4}{3}\right)}+\frac{\sqrt[3]{3}\pi x^4\mathrm{Gamma}\left(\frac{1}{3}\right)\mathrm{Gamma}\left(\frac{4}{3}\right)(\sqrt{3}\mathrm{AiryAiPrime}(x)-\mathrm{AiryBiPrime}(x)){}_1{F}_2(\frac{4}{3};\frac{5}{3},\frac{7}{3};\frac{x^3}{9})}{\mathrm{Gamma}\left(\frac{7}{3}\right)}+3\sqrt[6]{3}\pi x^2\mathrm{Gamma}\left(\frac{2}{3}\right)(3\mathrm{AiryAiPrime}(x)+\sqrt{3}\mathrm{AiryBiPrime}(x)){}_1{F}_2(\frac{2}{3};\frac{1}{3},\frac{5}{3};\frac{x^3}{9})+27\mathrm{Gamma}\left(\frac{1}{3}\right)\mathrm{Gamma}\left(\frac{5}{3}\right)(c_1\mathrm{AiryAiPrime}(x)+c_2\mathrm{AiryBiPrime}(x))}{\mathrm{Gamma}\left(\frac{5}{3}\right)}}{27\mathrm{Gamma}\left(\frac{1}{3}\right)}$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - x*y(x) + Derivative(y(x), (x, 2)) - Derivative(y(x), x)/x - 1/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**3 + x**2*y(x) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + 1 cannot be solved by the factorable group method