Internal problem ID [6819]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 87.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.578 (sec). Leaf size: 242
dsolve(x^2*(5+x+10*x^2)*diff(y(x),x$2)+x*(4+3*x+48*x^2)*diff(y(x),x)+(x+36*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {\sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}} \mathit {HG}\left (\frac {1-i \sqrt {199}}{1+i \sqrt {199}}, \frac {15721-179 i \sqrt {199}}{194275 i \sqrt {199}+641775}, -\frac {1}{5}, 0, \frac {4}{5}, -\frac {i \sqrt {199}}{995}, -\frac {20 x}{1+i \sqrt {199}}\right ) \left (i \sqrt {199}+20 x +1\right )^{-\frac {i \sqrt {199}}{1990}} \left (i \sqrt {199}-20 x -1\right )^{\frac {i \sqrt {199}}{1990}}}{10 x^{2}+x +5}+\frac {c_{2} {\mathrm e}^{-\frac {\sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}} \mathit {HG}\left (\frac {1-i \sqrt {199}}{1+i \sqrt {199}}, 0, 0, \frac {1}{5}, \frac {6}{5}, -\frac {i \sqrt {199}}{995}, -\frac {20 x}{1+i \sqrt {199}}\right ) x^{\frac {1}{5}} \left (i \sqrt {199}+20 x +1\right )^{-\frac {i \sqrt {199}}{1990}} \left (i \sqrt {199}-20 x -1\right )^{\frac {i \sqrt {199}}{1990}}}{10 x^{2}+x +5} \]
✓ Solution by Mathematica
Time used: 1.884 (sec). Leaf size: 88
DSolve[x^2*(5+x+10*x^2)*y''[x]+x*(4+3*x+48*x^2)*y'[x]+(x+36*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [5]{x} e^{-\frac {2 \text {ArcTan}\left (\frac {20 x+1}{\sqrt {199}}\right )}{5 \sqrt {199}}} \left (c_2 \int _1^x\frac {e^{\frac {2 \text {ArcTan}\left (\frac {20 K[1]+1}{\sqrt {199}}\right )}{5 \sqrt {199}}}}{K[1]^{6/5}}dK[1]+c_1\right )}{10 x^2+x+5} \\ \end{align*}