2.2.48 problem 47

Internal problem ID [8278]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 47
Date solved : Sunday, November 10, 2024 at 09:07:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-x y-x^{2}-\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 : 0.009 (sec)
Leaf size : 26

dsolve(diff(diff(y(x),x),x)-1/x*diff(y(x),x)-x*y(x)-x^2-1/x = 0, 
       y(x),singsol=all)
 

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

Mathematica DSolve solution

Solving time : 0.452 (sec)
Leaf size : 253

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

\[ y(x)\to \frac {\frac {3 \sqrt [6]{3} \pi \operatorname {Gamma}\left (-\frac {1}{3}\right ) \left (3 \operatorname {AiryAiPrime}(x)+\sqrt {3} \operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (-\frac {1}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{9}\right )}{x \operatorname {Gamma}\left (\frac {2}{3}\right )}+\frac {\frac {\sqrt [3]{3} \pi x \operatorname {Gamma}\left (\frac {1}{3}\right )^2 \left (\sqrt {3} \operatorname {AiryAiPrime}(x)-\operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {1}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {4}{3}\right )}+\frac {\sqrt [3]{3} \pi x^4 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right ) \left (\sqrt {3} \operatorname {AiryAiPrime}(x)-\operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {4}{3};\frac {5}{3},\frac {7}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {7}{3}\right )}+3 \sqrt [6]{3} \pi x^2 \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (3 \operatorname {AiryAiPrime}(x)+\sqrt {3} \operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {2}{3};\frac {1}{3},\frac {5}{3};\frac {x^3}{9}\right )+27 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) (c_1 \operatorname {AiryAiPrime}(x)+c_2 \operatorname {AiryBiPrime}(x))}{\operatorname {Gamma}\left (\frac {5}{3}\right )}}{27 \operatorname {Gamma}\left (\frac {1}{3}\right )} \]