Internal problem ID [10852]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 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]]
\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (-1+n \right ) y=0} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 47
dsolve(diff(y(x),x$2)+x*diff(y(x),x)+(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{2}} \operatorname {KummerM}\left (\frac {3}{2}-\frac {n}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) x +c_{2} {\mathrm e}^{-\frac {x^{2}}{2}} \operatorname {KummerU}\left (\frac {3}{2}-\frac {n}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 51
DSolve[y''[x]+x*y'[x]+(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {x^2}{2}} \left (c_1 \operatorname {HermiteH}\left (n-2,\frac {x}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (1-\frac {n}{2},\frac {1}{2},\frac {x^2}{2}\right )\right ) \]