59.1.25 problem 25

Internal problem ID [9197]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 25
Date solved : Sunday, March 30, 2025 at 02:25:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 69

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

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

✗ Sympy

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

False