33.24 problem 262

Internal problem ID [11085]
Internal file name [OUTPUT/10342_Wednesday_January_24_2024_10_18_09_PM_19808305/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: 262.
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.

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

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 an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
-> 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 a symmetry of the form [xi=0, eta=F(x)] 
   trying differential order: 2; exact nonlinear 
   trying symmetries linear in x and y(x) 
   <- linear symmetries successful`
 

✓ Solution by Maple

Time used: 1.0 (sec). Leaf size: 76

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

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

✗ Solution by Mathematica

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

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

Not solved