2.2.41 problem 40

Internal problem ID [8271]
Book : Own collection of miscellaneous problems
Section : section 2.0
Problem number : 40
Date solved : Sunday, November 10, 2024 at 09:05:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime }-x^{3} y-x^{4}&=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 : 0.015 (sec)
Leaf size : 32

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

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

Mathematica DSolve solution

Solving time : 0.191 (sec)
Leaf size : 219

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

\[ y(x)\to \frac {\sqrt [5]{-1} \operatorname {Gamma}\left (\frac {6}{5}\right ) \left (-5^{2/5} \sqrt [5]{x^{5/2}} x^{15/2} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {Hypergeometric0F1Regularized}\left (\frac {11}{5},\frac {x^5}{25}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{4/5} \left (x^{5/2}\right )^{4/5} \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{3/5} x^5 \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {6}{5},\frac {2 x^{5/2}}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )\right )}{25 x^{3/2} \text {Root}\left [25 \text {$\#$1}^5+1\&,5\right ]}+\frac {c_1 \sqrt {x} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )}{\sqrt [5]{5}}+\sqrt [5]{-\frac {1}{5}} c_2 \sqrt {x} \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right ) \]