55.34.7 problem 245

Internal problem ID [14018]
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 : 245
Date solved : Friday, October 03, 2025 at 07:23:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.363 (sec). Leaf size: 76

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

\[ y = \frac {\left (a x +b \right ) \left (c_2 \int \frac {{\mathrm e}^{\frac {b \left (n -3\right ) x^{2-n}+\left (-2+n \right ) \left (a \,x^{3-n}-2 \ln \left (x \right ) \left (n -3\right )\right )}{\left (-2+n \right ) \left (n -3\right )}} x^{2}}{\left (a x +b \right )^{2}}d x +c_1 \right )}{x} \]

✗ Mathematica

ode=x^n*D[y[x],{x,2}]+(2*x^(n-1)+a*x^2+b*x)*D[y[x],x]+b*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*y(x) + x**n*Derivative(y(x), (x, 2)) + (a*x**2 + b*x + 2*x**(n - 1))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-b*y(x) - x**n*Derivative(y(x), (x, 2)))/(a*x**2 + b*x + 2*x**(n - 1)) + Derivative(y(x), x) cannot be solved by the factorable group method