59.1.447 problem 461

Internal problem ID [9619]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 461
Date solved : Sunday, March 30, 2025 at 02:38:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x +1\right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

ode:=(2*x+1)*x*diff(diff(y(x),x),x)-2*(2*x^2-1)*diff(y(x),x)-4*(1+x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1}{x}+c_2 \,{\mathrm e}^{2 x} \]

✓ Mathematica. Time used: 0.242 (sec). Leaf size: 92

ode=(2*x+1)*x*D[y[x],{x,2}]-2*(2*x^2-1)*D[y[x],x]-4*(x+1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\left (\frac {1}{-2 K[1]-1}-1\right )dK[1]-\frac {1}{2} \int _1^x\frac {2-4 K[2]^2}{2 K[2]^2+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\left (\frac {1}{-2 K[1]-1}-1\right )dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(2*x + 1)*Derivative(y(x), (x, 2)) - (4*x + 4)*y(x) - (4*x**2 - 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False