2.2.42 problem 41

Internal problem ID [8272]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 41
Date solved : Sunday, November 10, 2024 at 09:07:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-x^{2} 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 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius 
   trying a solution in terms of MeijerG functions 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   <- Heun successful: received ODE is equivalent to the  HeunT  ODE, case  c = 0 
<- solving first the homogeneous part of the ODE successful`
 

Maple dsolve solution

Solving time : 2.695 (sec)
Leaf size : 55

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

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

Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

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

Not solved