55.29.49 problem 109

Internal problem ID [13882]
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 : 109
Date solved : Thursday, October 02, 2025 at 08:08:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 63

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

\[ y = \left (c_1 \int \frac {{\mathrm e}^{\int \frac {a \,x^{n}+b \,x^{m}+c -1}{x +\gamma }d x}}{x +\gamma }d x +c_2 \right ) {\mathrm e}^{-\int \frac {a \,x^{n}+b \,x^{m}+c -1}{x +\gamma }d x} \]

✗ Mathematica

ode=(x+\[Gamma])*D[y[x],{x,2}]+(a*x^n+b*x^m+c)*D[y[x],x]+(a*n*x^(n-1)+b*m*x^(m-1))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
Gamma = symbols("Gamma") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq((Gamma + x)*Derivative(y(x), (x, 2)) + (a*n*x**(n - 1) + b*m*x**(m - 1))*y(x) + (a*x**n + b*x**m + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Add object cannot be interpreted as an integer