2.46 problem 45

Internal problem ID [7182]
Internal file name [OUTPUT/6168_Sunday_June_05_2022_04_26_24_PM_50542001/index.tex]

Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 45.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-x^{2} y=x^{3}+x^{2}} \]

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`
 

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 57

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

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

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[y''[x]-x^2*y'[x]-x^2*y[x]-x^3-x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved