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, February 23, 2025 at 04:37:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-x y-x^{2}&=0 \end{align*}

Maple step by step solution

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`
 

Maple dsolve solution

Solving time : 0.010 (sec)
Leaf size : 44

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

\[ y = {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{2} +{\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{1} -\frac {x}{2} \]

✓Mathematica DSolve solution

Solving time : 0.318 (sec)
Leaf size : 224

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

\[ y(x)\to -\frac {e^{\frac {x^3}{6}} \left (12 \left (x^3\right )^{5/6} \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {7}{6}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (-\frac {2}{3};-\frac {1}{3};-\frac {x^3}{3}\right )+\sqrt [3]{2} 3^{2/3} \sqrt [6]{x^3} x^6 \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (\frac {2}{3};\frac {7}{3};-\frac {x^3}{3}\right )-4 \operatorname {Gamma}\left (\frac {7}{6}\right ) \left (6 \sqrt [3]{2} 3^{2/3} c_1 x^{5/2} \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right )+\operatorname {Gamma}\left (\frac {1}{6}\right ) \left (3 \left (x^3\right )^{5/6}+2 \sqrt [3]{-1} 3^{2/3} c_2 x^{5/2}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right )\right )\right )}{24\ 2^{2/3} 3^{5/6} x^2 \operatorname {Gamma}\left (\frac {7}{6}\right )} \]

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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