61.27.49 problem 59

Internal problem ID [12480]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 59
Date solved : Monday, March 31, 2025 at 05:36:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

TypeError : Add object cannot be interpreted as an integer