59.1.728 problem 745

Internal problem ID [9900]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 745
Date solved : Wednesday, March 05, 2025 at 08:00:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x -1\right )^{2} y^{\prime \prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 27

ode:=x*(x-1)^2*diff(diff(y(x),x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 \ln \left (x \right ) c_{2} x -c_{2} x^{2}+c_{1} x +c_{2}}{x -1} \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 62

ode=x*(x-1)^2*D[y[x],{x,2}]-2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x - 1)**2*Derivative(y(x), (x, 2)) - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x*(x - 1)**2*Derivative(y(x), (x, 2)) - 2*y(x) cannot be solved by the hypergeometric method