2.34.24 Problem 262

Internal problem ID [13922]
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
Date solved : Friday, December 19, 2025 at 08:46:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \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 \\ \end{align*}

2.34.24.1 ✓ Maple. Time used: 0.119 (sec). Leaf size: 76

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

\[ y = -\left (\int {\mathrm e}^{\int \frac {\lambda ^{3}-\lambda ^{2} x -\lambda \,x^{2}+x^{3}-2 a \,x^{n}-2 b \,x^{m}-2 c}{\left (a \,x^{n}+b \,x^{m}+c \right ) \left (-\lambda +x \right )}d x}d x c_1 +c_2 \right ) \left (\lambda -x \right ) \]

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 @ Moebi\ 
us 
-> 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
 

2.34.24.2 ✗ Mathematica

ode=(a*x^n+b*x^m+c)*D[y[x],{x,2}]+(\[Lambda]^2-x^2)*D[y[x],x]+(x+\[Lambda])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.34.24.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
lambda_ = symbols("lambda_") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq((lambda_ + x)*y(x) + (lambda_**2 - x**2)*Derivative(y(x), x) + (a*x**n + b*x**m + c)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Symbol object cannot be interpreted as an integer