2.2.52 Problem 51

Internal problem ID [10130]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 51
Date solved : Monday, December 01, 2025 at 08:40:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{3} y^{\prime }-x^{2} y-x^{3}&=0 \\ \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)-x^3*diff(y(x),x)-x^2*y(x)-x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\operatorname {KummerU}\left (\frac {1}{2}, \frac {5}{4}, \frac {x^{4}}{4}\right ) c_1 +\operatorname {KummerM}\left (\frac {1}{2}, \frac {5}{4}, \frac {x^{4}}{4}\right ) c_2 -\frac {1}{2}\right ) \]

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 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
      <- Kummer successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful
 

✓ Mathematica. Time used: 0.619 (sec). Leaf size: 337

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

\begin{align*} y(x)&\to \operatorname {Hypergeometric1F1}\left (\frac {1}{4},\frac {3}{4},\frac {x^4}{4}\right ) \int _1^x\frac {15 \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {K[1]^4}{4}\right ) K[1]^4}{5 \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {K[1]^4}{4}\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4},\frac {7}{4},\frac {K[1]^4}{4}\right ) K[1]^4-3 \operatorname {Hypergeometric1F1}\left (\frac {1}{4},\frac {3}{4},\frac {K[1]^4}{4}\right ) \left (2 \operatorname {Hypergeometric1F1}\left (\frac {3}{2},\frac {9}{4},\frac {K[1]^4}{4}\right ) K[1]^4+5 \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {K[1]^4}{4}\right )\right )}dK[1]+\frac {\sqrt [4]{-1} x \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {x^4}{4}\right ) \int _1^x\frac {(15-15 i) \operatorname {Hypergeometric1F1}\left (\frac {1}{4},\frac {3}{4},\frac {K[2]^4}{4}\right ) K[2]^3}{3 \operatorname {Hypergeometric1F1}\left (\frac {1}{4},\frac {3}{4},\frac {K[2]^4}{4}\right ) \left (2 \operatorname {Hypergeometric1F1}\left (\frac {3}{2},\frac {9}{4},\frac {K[2]^4}{4}\right ) K[2]^4+5 \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {K[2]^4}{4}\right )\right )-5 \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {K[2]^4}{4}\right ) \operatorname {Hypergeometric1F1}\left (\frac {5}{4},\frac {7}{4},\frac {K[2]^4}{4}\right ) K[2]^4}dK[2]}{\sqrt {2}}+c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{4},\frac {3}{4},\frac {x^4}{4}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) c_2 x \operatorname {Hypergeometric1F1}\left (\frac {1}{2},\frac {5}{4},\frac {x^4}{4}\right ) \end{align*}

✗ Sympy

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