59.1.546 problem 562

Internal problem ID [9718]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 562
Date solved : Sunday, March 30, 2025 at 02:45:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.441 (sec). Leaf size: 147

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

\[ y = \frac {{\mathrm e}^{-\frac {7 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}} x \left (c_1 \left (i \sqrt {3}-2 x -1\right )^{\frac {1}{4}+\frac {7 i \sqrt {3}}{12}} \left (i \sqrt {3}+2 x +1\right )^{\frac {1}{4}-\frac {7 i \sqrt {3}}{12}}+c_2 \left (i \sqrt {3}-2 x -1\right )^{-\frac {1}{4}-\frac {7 i \sqrt {3}}{12}} \left (i \sqrt {3}+2 x +1\right )^{\frac {3}{4}+\frac {7 i \sqrt {3}}{12}} \operatorname {hypergeom}\left (\left [1, \frac {1}{2}+\frac {7 i \sqrt {3}}{6}\right ], \left [\frac {3}{2}+\frac {7 i \sqrt {3}}{6}\right ], \frac {-i \sqrt {3}\, x +x +2}{i \sqrt {3}\, x +x +2}\right )\right )}{\left (x^{2}+x +1\right )^{{3}/{4}}} \]

✓ Mathematica. Time used: 0.552 (sec). Leaf size: 130

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

\[ y(x)\to \exp \left (\int _1^x\frac {1}{2} \left (\frac {K[1]-3}{K[1]^2+K[1]+1}+\frac {1}{K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\left (\frac {3 K[2]+5}{K[2]^2+K[2]+1}-\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {2 K[1]^2-2 K[1]+1}{2 K[1] \left (K[1]^2+K[1]+1\right )}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

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

False