Internal problem ID [7994]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 518.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 141
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)
\[ y \left (x \right ) = \frac {c_{1} \sqrt {x^{2}+x +1}\, \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {i \sqrt {3}}{6}}}{x^{2}}+\frac {c_{2} \sqrt {x^{2}+x +1}\, \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {i \sqrt {3}}{6}} \left (\int \frac {\left (\frac {i \sqrt {3}-2 x -1}{i \sqrt {3}+2 x +1}\right )^{-\frac {i \sqrt {3}}{6}}}{\left (x^{2}+x +1\right )^{\frac {3}{2}} \sqrt {x}}d x \right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.77 (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]
\[ y(x)\to \frac {\sqrt {x^2+x+1} e^{-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}} \left (c_2 \int _1^x\frac {e^{\frac {\arctan \left (\frac {2 K[1]+1}{\sqrt {3}}\right )}{\sqrt {3}}}}{\sqrt {K[1]} \left (K[1]^2+K[1]+1\right )^{3/2}}dK[1]+c_1\right )}{x^2} \]