Internal problem ID [6958]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 227.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+x y^{\prime }+\left (x +2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 37
dsolve(diff(y(x),x$2)+x*diff(y(x),x)+(2+x)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-x} \KummerM \left (\frac {3}{2}, \frac {1}{2}, -\frac {\left (x -2\right )^{2}}{2}\right )+c_{2} {\mathrm e}^{-x} \KummerU \left (\frac {3}{2}, \frac {1}{2}, -\frac {\left (x -2\right )^{2}}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.311 (sec). Leaf size: 84
DSolve[y''[x]+x*y'[x]+(2+x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} e^{-\frac {x^2}{2}+x-\frac {9}{2}} \left (e^{5/2} (x-3) (x-1) \left (\sqrt {2 \pi } c_2 \text {Erfi}\left (\frac {x-2}{\sqrt {2}}\right )+4 e^2 c_1\right )-2 c_2 e^{\frac {1}{2} (x-3)^2+x} (x-2)\right ) \\ \end{align*}