2.2.50 problem 49

Internal problem ID [8280]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 49
Date solved : Sunday, November 10, 2024 at 09:08:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{3} y-x^{4}-\frac {1}{x}&=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 : 0.008 (sec)
Leaf size : 26

dsolve(diff(diff(y(x),x),x)-1/x*diff(y(x),x)-y(x)*x^3-x^4-1/x = 0, 
       y(x),singsol=all)
 

\[ y = x \left (-1+\operatorname {BesselI}\left (\frac {2}{5}, \frac {2 x^{{5}/{2}}}{5}\right ) c_{2} +\operatorname {BesselK}\left (\frac {2}{5}, \frac {2 x^{{5}/{2}}}{5}\right ) c_{1} \right ) \]

Mathematica DSolve solution

Solving time : 0.334 (sec)
Leaf size : 316

DSolve[{D[y[x],{x,2}]-1/x*D[y[x],x]-x^3*y[x]-x^4-1/x==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\frac {5 \left (x^{5/2}\right )^{13/5} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {4}{5};\frac {3}{5},\frac {9}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {9}{5}\right )}-\frac {\sqrt [5]{5} \left (x^{5/2}\right )^{7/5} \operatorname {Gamma}\left (\frac {1}{5}\right ) \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {1}{5};\frac {6}{5},\frac {7}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {6}{5}\right )}+\frac {5 \left (x^{5/2}\right )^{3/5} \operatorname {Gamma}\left (-\frac {1}{5}\right ) \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (-\frac {1}{5};\frac {3}{5},\frac {4}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {4}{5}\right )}+\sqrt [5]{5} x^{5/2} \left (-\frac {x^5 \left (x^{5/2}\right )^{2/5} \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {6}{5};\frac {7}{5},\frac {11}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {11}{5}\right )}+10 \left (c_1 \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right )+(-1)^{2/5} c_2 \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right )\right )\right )}{10\ 5^{3/5} x^{3/2}} \]