53.2.5 problem 5

Internal problem ID [11298]
Book : Collection of Kovacic problems
Section : section 2. Solution found using all possible Kovacic cases
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 07:37:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

\[ y = {\mathrm e}^{x} \left (c_2 \,x^{3}+c_1 \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to \frac {1}{3} e^{x+1} \left (c_2 x^3+3 c_1\right ) \end{align*}

✗ Sympy

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

False