Internal problem ID [8291]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 819.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-x y^{\prime }-y x=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 77
dsolve(diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x} \left (x +2\right )+\frac {c_{2} \sqrt {2}\, \left (\pi \,{\mathrm e}^{-2} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) x -i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {1}{2} x^{2}+2 x}+2 \pi \,{\mathrm e}^{-2} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right )\right ) {\mathrm e}^{-x}}{2 \sqrt {\pi }} \]
✓ Solution by Mathematica
Time used: 0.113 (sec). Leaf size: 78
DSolve[y''[x]-x*y'[x]-x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{-x} \left (-\sqrt {2 \pi } c_2 \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 \sqrt {2} c_1 (x+2)+2 c_2 e^{\frac {1}{2} (x+2)^2}\right ) \]