59.1.545 problem 561

Internal problem ID [9717]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 561
Date solved : Sunday, March 30, 2025 at 02:45:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 1.453 (sec). Leaf size: 143

ode:=4*x^2*(x^2+x+1)*diff(diff(y(x),x),x)+12*x^2*(1+x)*diff(y(x),x)+(3*x^2+3*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sqrt {i \sqrt {3}-2 x -1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{2}} \sqrt {x}\, \left (c_1 \left (\frac {\sqrt {3}-2 i x -i}{\sqrt {3}+2 i x +i}\right )^{\frac {1}{4}-\frac {i \sqrt {3}}{4}}+c_2 \left (\frac {\sqrt {3}-2 i x -i}{\sqrt {3}+2 i x +i}\right )^{\frac {3}{4}+\frac {i \sqrt {3}}{4}} \operatorname {hypergeom}\left (\left [1, \frac {1}{2}+\frac {i \sqrt {3}}{2}\right ], \left [\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ], \frac {-i \sqrt {3}\, x +x +2}{i \sqrt {3}\, x +x +2}\right )\right )}{\left (x^{2}+x +1\right )^{{3}/{4}}} \]

✓ Mathematica. Time used: 0.467 (sec). Leaf size: 120

ode=4*x^2*(1+x+x^2)*D[y[x],{x,2}]+12*x^2*(1+x)*D[y[x],x]+(1+3*x+3*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*x**2*(x + 1)*Derivative(y(x), x) + 4*x**2*(x**2 + x + 1)*Derivative(y(x), (x, 2)) + (3*x**2 + 3*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False