61.27.44 problem 54

Internal problem ID [12475]
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 : 54
Date solved : Monday, March 31, 2025 at 05:35:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.450 (sec). Leaf size: 78

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

\[ y = -\left (c +x \right ) \left (\int \frac {{\mathrm e}^{-\frac {x^{n +1} \left (a \left (n +2\right ) \left (n +1\right ) x^{2}+\left (n +3\right ) \left (n +1\right ) \left (a c +b \right ) x +b c \left (n +3\right ) \left (n +2\right )\right )}{\left (n +1\right ) \left (n +2\right ) \left (n +3\right )}}}{\left (c +x \right )^{2}}d x c_1 +c_2 \right ) \]

✗ Mathematica

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

TypeError : Add object cannot be interpreted as an integer