Internal problem ID [6859]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 127.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (x +1\right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.809 (sec). Leaf size: 490
dsolve(4*x^2*(1+x+x^2)*diff(y(x),x$2)+12*x^2*(1+x)*diff(y(x),x)+(1+3*x+3*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (\frac {2 x +1+i \sqrt {3}}{i \sqrt {3}-2 x -1}\right )^{\frac {1}{4}-\frac {i \sqrt {3}}{4}} \sqrt {i \sqrt {3}-2 x -1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{2}} \hypergeom \left (\left [\frac {1}{2}+\frac {\sqrt {\frac {1-i \sqrt {3}}{1+i \sqrt {3}}}}{2}-\sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}, \frac {1}{2}-\frac {\sqrt {\frac {1-i \sqrt {3}}{1+i \sqrt {3}}}}{2}-\sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}\right ], \left [1-2 \sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}\right ], \frac {2 i \sqrt {3}\, x -2 x -4}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-2 x -1\right )}\right ) \sqrt {x}}{\left (x^{2}+x +1\right )^{\frac {3}{4}}}+\frac {c_{2} \left (\frac {2 x +1+i \sqrt {3}}{i \sqrt {3}-2 x -1}\right )^{\frac {1}{2}+\sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}} \sqrt {i \sqrt {3}-2 x -1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{2}} \hypergeom \left (\left [\frac {1}{2}-\frac {\sqrt {\frac {1-i \sqrt {3}}{1+i \sqrt {3}}}}{2}+\sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}, \frac {1}{2}+\frac {\sqrt {\frac {1-i \sqrt {3}}{1+i \sqrt {3}}}}{2}+\sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}\right ], \left [1+2 \sqrt {\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}\right )^{3}}}\right ], \frac {2 i \sqrt {3}\, x -2 x -4}{\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-2 x -1\right )}\right ) \sqrt {x}}{\left (x^{2}+x +1\right )^{\frac {3}{4}}} \]
✓ Solution by Mathematica
Time used: 1.119 (sec). Leaf size: 357
DSolve[4*x^2*(1+x+x^2)*y''[x]+12*x^2*(1+x)*y'[x]+(1+3*x+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1)^{5/6} \sqrt {x} e^{-\sqrt {3} \operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )} \left (\left (\sqrt {3}+3 i\right ) c_2 \sqrt {x^2+x+1} \left (e^{\sqrt {3} \operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}+e^{\left (\sqrt {3}+2 i\right ) \operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}\right ) \, _2F_1\left (1,\frac {1}{2} \left (1-i \sqrt {3}\right );\frac {1}{2} \left (3-i \sqrt {3}\right );\frac {i \sqrt {3} x+x+2}{-i \sqrt {3} x+x+2}\right )+c_2 \left (-2 \sqrt {3} \sqrt {x^2+x+1} e^{\sqrt {3} \operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}-2 \sqrt {3} \sqrt {x^2+x+1} e^{\left (\sqrt {3}+2 i\right ) \operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}+6 e^{\left (\sqrt {3}+i\right ) \operatorname {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}\right ) \, _2F_1\left (1,\frac {1}{2} \left (1-i \sqrt {3}\right );\frac {1}{2} \left (3-i \sqrt {3}\right );1-\frac {6 i}{2 \sqrt {3} x+\sqrt {3}+3 i}\right )-3 \left (\sqrt {3}+i\right ) c_1\right )}{6 \sqrt {x^2+x+1}} \\ \end{align*}