30.33 problem 181

Internal problem ID [11004]
Internal file name [OUTPUT/10261_Sunday_December_31_2023_11_33_25_AM_57004415/index.tex]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form \((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 181.
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.

\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y=0} \]

Maple trace Kovacic algorithm successful

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
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 
   A Liouvillian solution exists 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
   Solution has integrals. Trying a special function solution free of integrals... 
   -> 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 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   No special function solution was found. 
<- Kovacics algorithm successful`
 

✓ Solution by Maple

Time used: 0.437 (sec). Leaf size: 246

dsolve((a*x^2+b*x+c)*diff(y(x),x$2)+(x^3+k^3)*diff(y(x),x)-(x^2-k*x+k^2)*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \left (x +k \right ) \left (\left (\int \frac {\left (2 a x +b -\sqrt {-4 a c +b^{2}}\right )^{-\frac {k^{3}}{\sqrt {-4 a c +b^{2}}}} {\left (\frac {-2 a x -b +\sqrt {-4 a c +b^{2}}}{2 a x +\sqrt {-4 a c +b^{2}}+b}\right )}^{-\frac {3 b c}{2 a^{2} \sqrt {-4 a c +b^{2}}}} {\left (\frac {2 a x +\sqrt {-4 a c +b^{2}}+b}{-2 a x -b +\sqrt {-4 a c +b^{2}}}\right )}^{-\frac {b^{3}}{2 a^{3} \sqrt {-4 a c +b^{2}}}} \left (a \,x^{2}+b x +c \right )^{\frac {a c -b^{2}}{2 a^{3}}} {\left (\frac {2 a x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}\right )}^{\frac {k^{3}}{\sqrt {-4 a c +b^{2}}}} {\mathrm e}^{-\frac {x \left (a x -2 b \right )}{2 a^{2}}}}{\left (x +k \right )^{2}}d x \right ) c_{2} +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 3.224 (sec). Leaf size: 137

DSolve[(a*x^2+b*x+c)*y''[x]+(x^3+k^3)*y'[x]-(x^2-k*x+k^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {(k+x) \left (c_2 \int _1^x\frac {\exp \left (\frac {\left (b^3-3 a c b-2 a^3 k^3\right ) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{a^3 \sqrt {4 a c-b^2}}-\frac {K[1] (a K[1]-2 b)}{2 a^2}\right ) (c+K[1] (b+a K[1]))^{-\frac {b^2-a c}{2 a^3}}}{(k+K[1])^2}dK[1]+c_1\right )}{k} \]