55.34.5 problem 243

Internal problem ID [14016]
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 : 243
Date solved : Thursday, October 02, 2025 at 09:08:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.236 (sec). Leaf size: 56

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

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

✗ Mathematica

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

False