61.27.9 problem 19

Internal problem ID [12519]
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 : 19
Date solved : Tuesday, January 28, 2025 at 08:02:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+2 n y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.224 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 27

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

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