Internal problem ID [8037]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 561.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 143
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)
\[ y \left (x \right ) = c_{1} \sqrt {\frac {x}{x^{2}+x +1}}\, \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {i \sqrt {3}}{2}}+c_{2} \sqrt {\frac {x}{x^{2}+x +1}}\, \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {i \sqrt {3}}{2}} \left (\int \frac {\left (\frac {i \sqrt {3}-2 x -1}{i \sqrt {3}+2 x +1}\right )^{-\frac {i \sqrt {3}}{2}}}{x \sqrt {x^{2}+x +1}}d x \right ) \]
✓ Solution by Mathematica
Time used: 0.914 (sec). Leaf size: 93
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]
\[ y(x)\to \frac {\sqrt {x} e^{-\sqrt {3} \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )} \left (c_2 \int _1^x\frac {e^{\sqrt {3} \arctan \left (\frac {2 K[1]+1}{\sqrt {3}}\right )}}{K[1] \sqrt {K[1]^2+K[1]+1}}dK[1]+c_1\right )}{\sqrt {x^2+x+1}} \]