2.28.35 Problem 45

Internal problem ID [13706]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 45
Date solved : Friday, December 19, 2025 at 10:55:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{n -1} y&=0 \\ \end{align*}

2.28.35.1 ✓ Maple. Time used: 0.051 (sec). Leaf size: 96

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

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

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 Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Kummer successful 
<- special function solution successful
 

2.28.35.2 ✓ Mathematica. Time used: 0.061 (sec). Leaf size: 120

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

\begin{align*} y(x)&\to c_2 \left (\frac {1}{n}+1\right )^{-\frac {1}{n+1}} n^{-\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{n a+a},\frac {n+2}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{n a+a},\frac {n}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right ) \end{align*}

2.28.35.3 ✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer