61.27.8 problem 18

Internal problem ID [12518]
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
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 08:02:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+\left (n -1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.233 (sec). Leaf size: 104

dsolve(diff(y(x),x$2)+x*diff(y(x),x)+(n-1)*y(x)=0,y(x), singsol=all)
 

\[ y = -\frac {\left (-2 c_{1} n \left (-\frac {x^{2}}{2}+n +\frac {1}{2}\right ) \operatorname {KummerM}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+2 \left (-x^{2}+2 n +1\right ) c_{2} \operatorname {KummerU}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+n c_{1} \left (n +2\right ) \operatorname {KummerM}\left (-\frac {n}{2}-\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+4 \operatorname {KummerU}\left (-\frac {n}{2}-\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{2} \right ) {\mathrm e}^{-\frac {x^{2}}{2}} x}{n \left (n -1\right )} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 51

DSolve[D[y[x],{x,2}]+x*D[y[x],x]+(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {x^2}{2}} \left (c_1 \operatorname {HermiteH}\left (n-2,\frac {x}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (1-\frac {n}{2},\frac {1}{2},\frac {x^2}{2}\right )\right ) \]