61.33.25 problem 263

Internal problem ID [12684]
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 : 263
Date solved : Monday, March 31, 2025 at 06:51:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

TypeError : Add object cannot be interpreted as an integer