59.1.665 problem 682

Internal problem ID [9837]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 682
Date solved : Sunday, March 30, 2025 at 02:47:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 27

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

\[ y = \frac {c_2 \left (2+x \right ) \ln \left (1+x \right )+c_1 x +2 c_1 +4 c_2}{x} \]

✓ Mathematica. Time used: 0.451 (sec). Leaf size: 87

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + x**2*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)) - 2*y(x)/x**2 cannot be solved by the factorable group method