Internal problem ID [8038]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 562.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 137
dsolve(x^2*(1+x+x^2)*diff(y(x),x$2)-x*(1-4*x-2*x^2)*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} x \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {7 i \sqrt {3}}{6}}}{\sqrt {x^{2}+x +1}}+\frac {c_{2} x \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {7 i \sqrt {3}}{6}} \left (\int \frac {\left (\frac {i \sqrt {3}-2 x -1}{i \sqrt {3}+2 x +1}\right )^{-\frac {7 i \sqrt {3}}{6}}}{x \sqrt {x^{2}+x +1}}d x \right )}{\sqrt {x^{2}+x +1}} \]
✓ Solution by Mathematica
Time used: 0.914 (sec). Leaf size: 90
DSolve[x^2*(1+x+x^2)*y''[x]-x*(1-4*x-2*x^2)*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x e^{-\frac {7 \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}} \left (c_2 \int _1^x\frac {e^{\frac {7 \arctan \left (\frac {2 K[1]+1}{\sqrt {3}}\right )}{\sqrt {3}}}}{K[1] \sqrt {K[1]^2+K[1]+1}}dK[1]+c_1\right )}{\sqrt {x^2+x+1}} \]