59.1.542 problem 558

Internal problem ID [9714]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 558
Date solved : Sunday, March 30, 2025 at 02:44:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 x^{2} y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 39

ode:=2*x^2*(x+2)*diff(diff(y(x),x),x)+5*x^2*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\sqrt {2+x}\, \sqrt {2}\, c_2 -2 \,\operatorname {arctanh}\left (\frac {\sqrt {2+x}\, \sqrt {2}}{2}\right ) c_2 +c_1 \right ) \sqrt {x}}{\left (2+x \right )^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.392 (sec). Leaf size: 83

ode=2*x^2*(2+x)*D[y[x],{x,2}]+5*x^2*D[y[x],x]+(1+x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {K[1]+4}{4 K[1]^2+8 K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+4}{4 K[1]^2+8 K[1]}dK[1]\right )dK[2]+c_1\right )}{(x+2)^{5/4}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x + 2)*Derivative(y(x), (x, 2)) + 5*x**2*Derivative(y(x), x) + (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False