59.1.649 problem 666

Internal problem ID [9821]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 666
Date solved : Wednesday, March 05, 2025 at 07:59:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 78

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

\[ y = \left (c_{2} {\mathrm e}^{-\frac {\left (x -2\right )^{2}}{2}} \left (x -1\right ) \left (x -3\right ) \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-\left (x -2\right )^{2}}}{2}\right )-1\right ) \sqrt {\pi }-\sqrt {2}\, \sqrt {-\left (x -2\right )^{2}}\, c_{2} -c_{1} {\mathrm e}^{-\frac {\left (x -2\right )^{2}}{2}} \left (x -1\right ) \left (x -3\right )\right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.248 (sec). Leaf size: 63

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

\[ y(x)\to e^{x-\frac {x^2}{2}} \left (x^2-4 x+3\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2} (K[1]-4) K[1]}}{(K[1]-3)^2 (K[1]-1)^2}dK[1]+c_1\right ) \]

✗ Sympy

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

False