30.22 problem 170

Internal problem ID [10993]
Internal file name [OUTPUT/10250_Sunday_December_31_2023_11_24_11_AM_22104146/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: 170.
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.

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

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: 0.422 (sec). Leaf size: 1381

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

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 5.408 (sec). Leaf size: 104

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

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