61.31.7 problem 188

Internal problem ID [12609]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 188
Date solved : Monday, March 31, 2025 at 05:53:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+x \left (a \,x^{n}+b \right ) y^{\prime }-\left (a \,x^{n}-a b \,x^{n -1}+b \right ) y&=0 \end{align*}

✗ Maple

ode:=x^3*diff(diff(y(x),x),x)+x*(a*x^n+b)*diff(y(x),x)-(a*x^n-a*b*x^(n-1)+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=x^3*D[y[x],{x,2}]+x*(a*x^n+b)*D[y[x],x]-(a*x^n-a*b*x^(n-1)+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(x**3*Derivative(y(x), (x, 2)) + x*(a*x**n + b)*Derivative(y(x), x) - (-a*b*x**(n - 1) + a*x**n + b)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*b*x**(n - 1)*y(x) + a*x**n*y(x) + b*y(x) - x**3*Derivative(y(x), (x, 2)))/(x*(a*x**n + b)) cannot be solved by the factorable group method