Internal problem ID [10842]
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]]
\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (n -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 104
dsolve(diff(y(x),x$2)+x*diff(y(x),x)+(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-2 \left (-\frac {x^{2}}{2}+n +\frac {1}{2}\right ) c_{1} n \operatorname {KummerM}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+2 \left (-x^{2}+2 n +1\right ) c_{2} \operatorname {KummerU}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+n c_{1} \left (n +2\right ) \operatorname {KummerM}\left (-\frac {n}{2}-\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+4 \operatorname {KummerU}\left (-\frac {n}{2}-\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{2} \right ) x \,{\mathrm e}^{-\frac {x^{2}}{2}}}{n \left (n -1\right )} \]
✓ 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]
\[ y(x)\to e^{-\frac {x^2}{2}} \left (c_1 \operatorname {HermiteH}\left (n-2,\frac {x}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (1-\frac {n}{2},\frac {1}{2},\frac {x^2}{2}\right )\right ) \]