Internal problem ID [7559]
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]]
\[ \boxed {\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 143
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)
\[ y \left (x \right ) = \frac {c_{1} \left (\frac {i \sqrt {11}-6 x -1}{i \sqrt {11}+6 x +1}\right )^{-\frac {i \sqrt {11}}{22}} x}{\left (3 x^{2}+x +1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (\frac {i \sqrt {11}-6 x -1}{i \sqrt {11}+6 x +1}\right )^{-\frac {i \sqrt {11}}{22}} x \left (\int \frac {\sqrt {3 x^{2}+x +1}\, \left (\frac {i \sqrt {11}+6 x +1}{i \sqrt {11}-6 x -1}\right )^{-\frac {i \sqrt {11}}{22}}}{x^{2}}d x \right )}{\left (3 x^{2}+x +1\right )^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 3.347 (sec). Leaf size: 93
DSolve[(1+x+3*x^2)*y''[x]+(2+15*x)*y'[x]+12*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x e^{\frac {\arctan \left (\frac {6 x+1}{\sqrt {11}}\right )}{\sqrt {11}}} \left (c_2 \int _1^x\frac {e^{-\frac {\arctan \left (\frac {6 K[1]+1}{\sqrt {11}}\right )}{\sqrt {11}}} \sqrt {3 K[1]^2+K[1]+1}}{K[1]^2}dK[1]+c_1\right )}{\left (3 x^2+x+1\right )^{3/2}} \]