53.1.314 problem 319

Internal problem ID [10786]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 319
Date solved : Tuesday, September 30, 2025 at 07:31:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 17

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

\[ y = c_1 x +c_2 \,{\mathrm e}^{-x} \left (x +4\right ) \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 96

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

\begin{align*} y(x)&\to \frac {2 i \left ((c_1 (x+2)+2 i c_2) \cosh \left (\frac {x+2}{2}\right )-(i c_2 (x+2)+2 c_1) \sinh \left (\frac {x+2}{2}\right )\right ) \exp \left (\int _1^x\frac {1-K[1]}{2 K[1]+4}dK[1]\right )}{\sqrt {\pi } (-i (x+2))^{3/2}} \end{align*}

✗ Sympy

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

False