61.28.40 problem 100

Internal problem ID [12521]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 100
Date solved : Monday, March 31, 2025 at 05:37:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.197 (sec). Leaf size: 53

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

False