30.21 problem 169

Internal problem ID [10992]
Internal file name [OUTPUT/10249_Sunday_December_31_2023_11_24_09_AM_34106090/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: 169.
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.

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

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 
      <- Heun successful: received ODE is equivalent to the  HeunC  ODE, case  a <> 0, e <> 0, c = 0 
   <- Kovacics algorithm successful`
 

Solution by Maple

Time used: 0.281 (sec). Leaf size: 939

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

\[ y \left (x \right ) = \left (-a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} \left (c_{2} \left (a x +\sqrt {-a b}\right )^{-\frac {-2 a^{2} b +\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}} \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, -\frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {\lambda d}{a}-\frac {b c \lambda }{a^{2}}+\frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) {\mathrm e}^{\frac {-i \pi \sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}+i \pi \sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-4 b \left (a^{2} \left (\frac {d}{\sqrt {b}\, \sqrt {a}}-\frac {\sqrt {b}\, c}{a^{\frac {3}{2}}}\right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )+\left (-2 a \lambda +c \right ) \sqrt {-a b}-2 a x \left (a \lambda -c \right )\right )}{8 a^{2} b}}+c_{1} \left (a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}} {\mathrm e}^{x \lambda +\frac {\sqrt {-a b}\, \lambda }{a}-\frac {c x}{a}-\frac {\sqrt {-a b}\, c}{2 a^{2}}-\frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d}{2 \sqrt {a}\, \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c}{2 a^{\frac {3}{2}}}} \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {\lambda d}{a}-\frac {b c \lambda }{a^{2}}+\frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right )\right ) \]

Solution by Mathematica

Time used: 2.859 (sec). Leaf size: 74

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

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