Internal problem ID [7114]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 385.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.975 (sec). Leaf size: 362
dsolve(2*x^2*(1+x+x^2)*diff(y(x), x$2) + x*(9+11*x+11*x^2)*diff(y(x), x) + (6+10*x+7*x^2)*y(x) = 0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}} \mathit {HG}\left (\frac {1-i \sqrt {3}}{1+i \sqrt {3}}, 0, 0, \frac {5}{2}, \frac {1}{2}, \frac {5 i \sqrt {3}-3}{-3+3 i \sqrt {3}}, -\frac {2 x}{1+i \sqrt {3}}\right ) \left (2 x +1+i \sqrt {3}\right )^{\frac {5 i \sqrt {3}-3}{-6+6 i \sqrt {3}}} \left (i \sqrt {3}-2 x -1\right )^{\frac {-16 i \sqrt {3}-592}{\left (1+i \sqrt {3}\right )^{3} \left (-1+i \sqrt {3}\right )^{2} \left (13 i \sqrt {3}-9\right )}}}{\left (x^{2}+x +1\right )^{\frac {1}{4}} x^{2}}+\frac {c_{2} {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}} \mathit {HG}\left (\frac {1-i \sqrt {3}}{1+i \sqrt {3}}, \frac {16}{\left (1+i \sqrt {3}\right )^{3} \left (-1+i \sqrt {3}\right )^{2}}, \frac {1}{2}, 3, \frac {3}{2}, \frac {5 i \sqrt {3}-3}{-3+3 i \sqrt {3}}, -\frac {2 x}{1+i \sqrt {3}}\right ) \left (2 x +1+i \sqrt {3}\right )^{\frac {5 i \sqrt {3}-3}{-6+6 i \sqrt {3}}} \left (i \sqrt {3}-2 x -1\right )^{\frac {-16 i \sqrt {3}-592}{\left (1+i \sqrt {3}\right )^{3} \left (-1+i \sqrt {3}\right )^{2} \left (13 i \sqrt {3}-9\right )}}}{\left (x^{2}+x +1\right )^{\frac {1}{4}} x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 1.668 (sec). Leaf size: 93
DSolve[2*x^2*(1+x+x^2)*y''[x] + x*(9+11*x+11*x^2)*y'[x] + (6+10*x+7*x^2)*y[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {x^2+x+1} e^{-\frac {\operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}} \left (c_2 \int _1^x\frac {e^{\frac {\operatorname {ArcTan}\left (\frac {2 K[1]+1}{\sqrt {3}}\right )}{\sqrt {3}}}}{\sqrt {K[1]} \left (K[1]^2+K[1]+1\right )^{3/2}}dK[1]+c_1\right )}{x^2} \\ \end{align*}