61.28.44 problem 104

Internal problem ID [12525]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 104
Date solved : Monday, March 31, 2025 at 05:37:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

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