33.3 problem 241

Internal problem ID [11064]
Internal file name [OUTPUT/10321_Wednesday_January_24_2024_10_07_12_PM_56094527/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 2.1.2-8. Other equations.
Problem number: 241.
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.

\[ \boxed {x^{n} y^{\prime \prime }+c \left (a x +b \right )^{n -4} y=0} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Whittaker 
      -> 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: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
<- unable to find a useful change of variables 
   trying differential order: 2; exact nonlinear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   trying to convert to an ODE of Bessel type 
   -> trying reduction of order to Riccati 
      trying Riccati sub-methods: 
         trying Riccati_symmetries 
         -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
         -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
         -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
--- Trying Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3`[0, y]
 

✗ Solution by Maple

dsolve(x^n*diff(y(x),x$2)+c*(a*x+b)^(n-4)*y(x)=0,y(x), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[x^n*y''[x]+c*(a*x+b)^(n-4)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved