2.30.29 Problem 138

Internal problem ID [13799]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 138
Date solved : Friday, December 19, 2025 at 12:46:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (k \left (a -k \right ) x^{2}+\left (a n +b k -2 n k \right ) x +n \left (-n +b -1\right )\right ) y&=0 \\ \end{align*}

2.30.29.1 ✓ Maple. Time used: 0.004 (sec). Leaf size: 48

ode:=x^2*diff(diff(y(x),x),x)+(a*x^2+b*x)*diff(y(x),x)+(k*(a-k)*x^2+(a*n+b*k-2*k*n)*x+n*(b-n-1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{-n} {\mathrm e}^{-k x}+c_2 \operatorname {WhittakerM}\left (-\frac {b}{2}+n , -\frac {b}{2}+n +\frac {1}{2}, \left (-2 k +a \right ) x \right ) x^{-\frac {b}{2}} {\mathrm e}^{-\frac {a x}{2}} \]

Maple trace

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 
<- Kovacics algorithm successful
 

2.30.29.2 ✓ Mathematica. Time used: 0.315 (sec). Leaf size: 64

ode=x^2*D[y[x],{x,2}]+(a*x^2+b*x)*D[y[x],x]+(k*(a-k)*x^2+(a*n+b*k-2*k*n)*x+n*(b-n-1))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-k x} x^{-n} \left (c_1-c_2 x^{-b+2 n+1} (x (a-2 k))^{b-2 n-1} \Gamma (-b+2 n+1,(a-2 k) x)\right ) \end{align*}

2.30.29.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
k = symbols("k") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (a*x**2 + b*x)*Derivative(y(x), x) + (k*x**2*(a - k) + n*(b - n - 1) + x*(a*n + b*k - 2*k*n))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_regular')