59.1.503 problem 519

Internal problem ID [9675]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 519
Date solved : Wednesday, March 05, 2025 at 07:57:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y&=0 \end{align*}

✓ Maple. Time used: 1.974 (sec). Leaf size: 38

ode:=3*x^2*diff(diff(y(x),x),x)+2*x*(-2*x^2+x+1)*diff(y(x),x)+(-8*x^2+2*x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_{1} x^{{1}/{3}} {\mathrm e}^{\frac {2 \left (x -1\right ) x}{3}}+c_{2} \operatorname {HeunB}\left (-\frac {1}{3}, \frac {\sqrt {6}}{3}, -\frac {7}{3}, \frac {4 \sqrt {6}}{9}, -\frac {\sqrt {6}\, x}{3}\right ) \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 53

ode=3*x^2*D[y[x],{x,2}]+2*x*(1+x-2*x^2)*D[y[x],x]+(2*x-8*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\frac {2}{3} (x-1) x} \sqrt [3]{x} \left (c_2 \int _1^x\frac {e^{-\frac {2}{3} (K[1]-1) K[1]}}{K[1]^{4/3}}dK[1]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + 2*x*(-2*x**2 + x + 1)*Derivative(y(x), x) + (-8*x**2 + 2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False