61.33.20 problem 258

Internal problem ID [12679]
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 : 258
Date solved : Monday, March 31, 2025 at 06:50:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

TypeError : Symbol object cannot be interpreted as an integer