59.1.798 problem 821

Internal problem ID [9970]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 821
Date solved : Sunday, March 30, 2025 at 02:50:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 48

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

\[ y = {\mathrm e}^{-x} \left (\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +2\right )}{2}\right ) c_2 \left (x +2\right ) {\mathrm e}^{-2}-i \sqrt {2}\, \sqrt {\pi }\, c_2 \,{\mathrm e}^{\frac {x \left (x +4\right )}{2}}+c_1 \left (x +2\right )\right ) \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 78

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

\[ y(x)\to \frac {1}{2} e^{-x} \left (-\sqrt {2 \pi } c_2 \sqrt {(x+2)^2} \text {erfi}\left (\frac {\sqrt {(x+2)^2}}{\sqrt {2}}\right )+2 \sqrt {2} c_1 (x+2)+2 c_2 e^{\frac {1}{2} (x+2)^2}\right ) \]

✗ Sympy

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

False