27.9 problem 19

Internal problem ID [10100]

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: 19.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {y^{\prime \prime }-2 x y^{\prime }+2 y n=0} \end {gather*}

Solution by Maple

Time used: 0.067 (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)
 

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

Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 27

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

\begin{align*} y(x)\to c_1 \text {HermiteH}(n,x)+c_2 \, _1F_1\left (-\frac {n}{2};\frac {1}{2};x^2\right ) \\ \end{align*}