2.2.50 Problem 49

Internal problem ID [8854]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 49
Date solved : Sunday, March 30, 2025 at 01:42:58 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^3*y(x)-x^4-1/x = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=x(-1+c_2\mathrm{BesselI}(\frac{2}{5},\frac{2x^{5/2}}{5})+c_1\mathrm{BesselK}(\frac{2}{5},\frac{2x^{5/2}}{5}))$$

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: 0.334 (sec). Leaf size: 316

ode=D[y[x],{x,2}]-1/x*D[y[x],x]-x^3*y[x]-x^4-1/x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to \frac{\frac{5{\left(x^{5/2}\right)}^{13/5}\mathrm{Gamma}\left(\frac{4}{5}\right)\mathrm{Gamma}\left(\frac{7}{5}\right)\mathrm{BesselI}(\frac{2}{5},\frac{2x^{5/2}}{5}){}_1{F}_2(\frac{4}{5};\frac{3}{5},\frac{9}{5};\frac{x^5}{25})}{\mathrm{Gamma}\left(\frac{9}{5}\right)}-\frac{\sqrt[5]{5}{\left(x^{5/2}\right)}^{7/5}\mathrm{Gamma}\left(\frac{1}{5}\right)\mathrm{Gamma}\left(\frac{3}{5}\right)\mathrm{BesselI}(-\frac{2}{5},\frac{2x^{5/2}}{5}){}_1{F}_2(\frac{1}{5};\frac{6}{5},\frac{7}{5};\frac{x^5}{25})}{\mathrm{Gamma}\left(\frac{6}{5}\right)}+\frac{5{\left(x^{5/2}\right)}^{3/5}\mathrm{Gamma}(-\frac{1}{5})\mathrm{Gamma}\left(\frac{7}{5}\right)\mathrm{BesselI}(\frac{2}{5},\frac{2x^{5/2}}{5}){}_1{F}_2(-\frac{1}{5};\frac{3}{5},\frac{4}{5};\frac{x^5}{25})}{\mathrm{Gamma}\left(\frac{4}{5}\right)}+\sqrt[5]{5}{x}^{5/2}(-\frac{x^5{\left(x^{5/2}\right)}^{2/5}\mathrm{Gamma}\left(\frac{3}{5}\right)\mathrm{Gamma}\left(\frac{6}{5}\right)\mathrm{BesselI}(-\frac{2}{5},\frac{2x^{5/2}}{5}){}_1{F}_2(\frac{6}{5};\frac{7}{5},\frac{11}{5};\frac{x^5}{25})}{\mathrm{Gamma}\left(\frac{11}{5}\right)}+10(c_1\mathrm{Gamma}\left(\frac{3}{5}\right)\mathrm{BesselI}(-\frac{2}{5},\frac{2x^{5/2}}{5})+(-1{)}^{2/5}{c}_2\mathrm{Gamma}\left(\frac{7}{5}\right)\mathrm{BesselI}(\frac{2}{5},\frac{2x^{5/2}}{5})))}{105^{3/5}{x}^{3/2}}$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**4 - x**3*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**5 + x**4*y(x) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + 1 cannot be solved by the factorable group method