2.2.50 Problem 49

Internal problem ID [10128]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 49
Date solved : Monday, December 01, 2025 at 08:39:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{3} y-x^{4}-\frac {1}{x}&=0 \\ \end{align*}

✓ Maple. Time used: 0.003 (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);
 

\[ y = x \left (-1+c_2 \operatorname {BesselI}\left (\frac {2}{5}, \frac {2 x^{{5}/{2}}}{5}\right )+c_1 \operatorname {BesselK}\left (\frac {2}{5}, \frac {2 x^{{5}/{2}}}{5}\right )\right ) \]

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: 0.201 (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]
 

\begin{align*} y(x)&\to \frac {\frac {5 \left (x^{5/2}\right )^{13/5} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {4}{5};\frac {3}{5},\frac {9}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {9}{5}\right )}-\frac {\sqrt [5]{5} \left (x^{5/2}\right )^{7/5} \operatorname {Gamma}\left (\frac {1}{5}\right ) \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {1}{5};\frac {6}{5},\frac {7}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {6}{5}\right )}+\frac {5 \left (x^{5/2}\right )^{3/5} \operatorname {Gamma}\left (-\frac {1}{5}\right ) \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (-\frac {1}{5};\frac {3}{5},\frac {4}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {4}{5}\right )}+\sqrt [5]{5} x^{5/2} \left (-\frac {x^5 \left (x^{5/2}\right )^{2/5} \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {6}{5};\frac {7}{5},\frac {11}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {11}{5}\right )}+10 \left (c_1 \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right )+(-1)^{2/5} c_2 \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right )\right )\right )}{10\ 5^{3/5} x^{3/2}} \end{align*}

✗ 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