Internal problem ID [6803]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 71.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.322 (sec). Leaf size: 305
dsolve((1+x+3*x^2)*diff(y(x),x$2)+(2+15*x)*diff(y(x),x)+12*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (i \sqrt {11}-6 x -1\right )^{\frac {3}{2}} {\mathrm e}^{\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{22}} \hypergeom \left (\left [\frac {\sqrt {1078+66 i \sqrt {11}}}{44}+\frac {1}{2}-\frac {\sqrt {1078-66 i \sqrt {11}}}{44}, \frac {\sqrt {1078+66 i \sqrt {11}}}{44}+\frac {1}{2}-\frac {\sqrt {1078-66 i \sqrt {11}}}{44}\right ], \left [1-\frac {\sqrt {1078-66 i \sqrt {11}}}{22}\right ], -\frac {i \sqrt {11}\, \left (i \sqrt {11}+6 x +1\right )}{22}\right ) \left (-36 x^{2}-12 x -12\right )^{\frac {i \sqrt {11}}{44}}}{\left (3 x^{2}+x +1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (i \sqrt {11}+6 x +1\right )^{\frac {5}{4}-\frac {i \sqrt {11}}{44}} \left (i \sqrt {11}-6 x -1\right )^{\frac {5}{4}+\frac {i \sqrt {11}}{44}} {\mathrm e}^{\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{22}} \hypergeom \left (\left [\frac {\sqrt {1078+66 i \sqrt {11}}}{44}+\frac {1}{2}+\frac {\sqrt {1078-66 i \sqrt {11}}}{44}, \frac {\sqrt {1078+66 i \sqrt {11}}}{44}+\frac {1}{2}+\frac {\sqrt {1078-66 i \sqrt {11}}}{44}\right ], \left [1+\frac {\sqrt {1078-66 i \sqrt {11}}}{22}\right ], -\frac {i \sqrt {11}\, \left (i \sqrt {11}+6 x +1\right )}{22}\right )}{\left (3 x^{2}+x +1\right )^{\frac {5}{4}}} \]
✓ Solution by Mathematica
Time used: 2.36 (sec). Leaf size: 181
DSolve[(1+x+3*x^2)*y''[x]+(2+15*x)*y'[x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 12 \sqrt {2} \sqrt [4]{3} \left (-6 i x+\sqrt {11}-i\right )^{\frac {1}{44} i \left (\sqrt {11}+66 i\right )} \left (6 i x+\sqrt {11}+i\right )^{-\frac {3}{2}-\frac {i}{4 \sqrt {11}}} x e^{\frac {\operatorname {ArcTan}\left (\frac {6 x+1}{\sqrt {11}}\right )}{2 \sqrt {11}}} \left (c_2 \int _1^x\frac {\left (-6 i K[1]+\sqrt {11}-i\right )^{\frac {1}{2}-\frac {i}{2 \sqrt {11}}} \left (6 i K[1]+\sqrt {11}+i\right )^{\frac {1}{2}+\frac {i}{2 \sqrt {11}}}}{K[1]^2}dK[1]+c_1\right ) \\ \end{align*}