59.1.57 problem 59

Internal problem ID [9229]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 59
Date solved : Wednesday, March 05, 2025 at 07:38:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

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

\[ y = -x \left (c_{2} \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) \pi -c_{1} \right ) {\mathrm e}^{-\frac {x^{2}}{2}}+i \sqrt {\pi }\, \sqrt {2}\, c_{2} \]

✓ Mathematica. Time used: 0.048 (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