55.28.9 problem 19

Internal problem ID [13792]
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 : Friday, October 03, 2025 at 06:54:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 31

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

\[ y = x \left (\operatorname {KummerU}\left (\frac {1}{2}-\frac {n}{2}, \frac {3}{2}, x^{2}\right ) c_2 +\operatorname {KummerM}\left (\frac {1}{2}-\frac {n}{2}, \frac {3}{2}, x^{2}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 27

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

\begin{align*} y(x)&\to c_1 \operatorname {HermiteH}(n,x)+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {n}{2},\frac {1}{2},x^2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(2*n*y(x) - 2*x*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False