55.28.36 problem 46

Internal problem ID [13819]
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 : 46
Date solved : Thursday, October 02, 2025 at 08:07:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to (c_2 x+c_1) e^{-\frac {a x^{n+1}}{n+1}} \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded