2.47 problem 46

Internal problem ID [7183]
Internal file name [OUTPUT/6169_Sunday_June_05_2022_04_26_27_PM_63707127/index.tex]

Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 46.
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.

\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-x^{3} y=x^{4}+x^{2}} \]

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: 0.031 (sec). Leaf size: 74

dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)-x^3*y(x)-x^4-x^2=0,y(x), singsol=all)
 

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

✗ Solution by Mathematica

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

DSolve[y''[x]-x^2*y'[x]-x^3*y[x]-x^4-x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved