Internal problem ID [7896]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 416.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-3 y x=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 76
dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)-3*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{3}}{3}} x -\frac {9 c_{2} {\mathrm e}^{\frac {x^{3}}{3}} 3^{\frac {2}{3}} {\mathrm e}^{-\frac {x^{3}}{6}} \left (x^{6} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )+5 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right ) x^{3}+10 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )\right )}{10 x^{3} \left (x^{3}\right )^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.08 (sec). Leaf size: 51
DSolve[y''[x]-x^2*y'[x]-3*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{9} e^{\frac {x^3}{3}} \left (9 c_1 x-3^{2/3} c_2 \sqrt [3]{x^3} \Gamma \left (-\frac {1}{3},\frac {x^3}{3}\right )\right ) \]