Internal problem ID [10099]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-1 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 18.
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: 0.016 (sec). Leaf size: 47
dsolve(diff(y(x),x$2)+x*diff(y(x),x)+(n-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{2}} \KummerM \left (\frac {3}{2}-\frac {n}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) x +c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \KummerU \left (\frac {3}{2}-\frac {n}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 51
DSolve[y''[x]+x*y'[x]+(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {x^2}{2}} \left (c_1 H_{n-2}\left (\frac {x}{\sqrt {2}}\right )+c_2 \, _1F_1\left (1-\frac {n}{2};\frac {1}{2};\frac {x^2}{2}\right )\right ) \\ \end{align*}