61.31.27 problem 208

Internal problem ID [12629]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 208
Date solved : Wednesday, March 05, 2025 at 07:49:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (\alpha x +\beta \right ) y&=0 \end{align*}

✗ Maple

ode:=(a*x^3+b*x^2+c*x+d)*diff(diff(y(x),x),x)+(alpha*x^2+(alpha*gamma+beta)*x+beta*lambda)*diff(y(x),x)-(alpha*x+beta)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=(a*x^3+b*x^2+c*x+d)*D[y[x],{x,2}]+(\[Alpha]*x^2+(\[Alpha]*\[Gamma]+\[Beta])*x+\[Beta]*\[Lambda])*D[y[x],x]-(\[Alpha]*x+\[Beta])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✗ Sympy

from sympy import * 
x = symbols("x") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
Gamma = symbols("Gamma") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
cg = symbols("cg") 
y = Function("y") 
ode = Eq((-Alpha*x - BETA)*y(x) + (Alpha*x**2 + BETA*cg + x*(Alpha*Gamma + BETA))*Derivative(y(x), x) + (a*x**3 + b*x**2 + c*x + d)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False