Internal problem ID [8621]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1041.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime } x +\left (n +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 47
dsolve(diff(diff(y(x),x),x)+x*diff(y(x),x)+(n+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{2}} \KummerM \left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) x +c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \KummerU \left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 47
DSolve[(1 + n)*y[x] + x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {x^2}{2}} \left (c_1 \text {HermiteH}\left (n,\frac {x}{\sqrt {2}}\right )+c_2 \, _1F_1\left (-\frac {n}{2};\frac {1}{2};\frac {x^2}{2}\right )\right ) \\ \end{align*}