Internal problem ID [7979]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 503.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+y^{\prime } x^{5}+6 y x^{4}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 81
dsolve(diff(y(x),x$2)+x^5*diff(y(x),x)+6*x^4*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{6}}{6}} x -\frac {2 c_{2} {\mathrm e}^{-\frac {x^{6}}{6}} \left (6^{\frac {2}{3}} \sqrt {3}\, \sqrt {2}\, \left (-x^{6}\right )^{\frac {5}{6}} {\mathrm e}^{\frac {x^{6}}{6}}+6 \Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right ) x^{6}-6 \Gamma \left (\frac {5}{6}\right ) x^{6}\right )}{\left (-x^{6}\right )^{\frac {5}{6}} \left (\sqrt {3}+i\right )} \]
✓ Solution by Mathematica
Time used: 0.117 (sec). Leaf size: 53
DSolve[y''[x]+x^5*y'[x]+6*x^4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{36} e^{-\frac {x^6}{6}} \left (36 c_1 x-6^{5/6} c_2 \sqrt [6]{-x^6} \Gamma \left (-\frac {1}{6},-\frac {x^6}{6}\right )\right ) \]