53.1.669 problem 686

Internal problem ID [11141]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 686
Date solved : Tuesday, September 30, 2025 at 07:36:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to -\frac {2 e^{\frac {x+1}{2}} \left ((c_1 x+2 i c_2) \cosh \left (\frac {x}{2}\right )-(i c_2 x+2 c_1) \sinh \left (\frac {x}{2}\right )\right )}{\sqrt {\pi } \sqrt {-i x} \sqrt {x}} \end{align*}

✗ Sympy

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

False