61.33.12 problem 250

Internal problem ID [12671]
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 : 250
Date solved : Wednesday, March 05, 2025 at 08:15:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

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

\[ y = \left (\left (\int \frac {1}{\left (a \,x^{n}+b x +c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \left (a \,x^{n}+b x +c \right ) \]

✗ Mathematica

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

Not solved

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer