53.1.219 problem 222

Internal problem ID [10691]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 222
Date solved : Tuesday, September 30, 2025 at 07:31:02 PM
CAS classification : [_erf]

\begin{align*} y^{\prime \prime }+2 x y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)+2*x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 51

ode=D[y[x],{x,2}]+2*x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x^2} \left (-\sqrt {\pi } c_2 \sqrt {x^2} \text {erfi}\left (\sqrt {x^2}\right )+c_2 e^{x^2}+2 c_1 x\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False