53.2.8 problem 8

Internal problem ID [11301]
Book : Collection of Kovacic problems
Section : section 2. Solution found using all possible Kovacic cases
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 07:37:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

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

\[ y = \ln \left (x \right ) c_2 x +c_1 x +c_2 \]

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 75

ode=(x^2-x)*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 \sqrt {1-x} \exp \left (\int _1^x\left (\frac {1}{K[1]}+\frac {1}{2-2 K[1]}\right )dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\left (\frac {1}{K[1]}+\frac {1}{2-2 K[1]}\right )dK[1]\right )dK[2]+c_1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + Derivative(y(x), (x, 2)) - y(x)/x cannot be solved by the factorable group method