59.1.146 problem 148

Internal problem ID [9318]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 148
Date solved : Wednesday, March 05, 2025 at 07:47:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.220 (sec). Leaf size: 102

ode:=3*x^2*(x^2+3)*diff(diff(y(x),x),x)+x*(11*x^2+3)*diff(y(x),x)+(5*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{{1}/{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (9 x^{2}+27\right )^{{1}/{3}} \sqrt {3}}{6+\left (9 x^{2}+27\right )^{{1}/{3}}}\right ) c_{2} +3 \,3^{{1}/{3}} c_{1} +2 \ln \left (3-\left (9 x^{2}+27\right )^{{1}/{3}}\right ) c_{2} -\ln \left (\left (9 x^{2}+27\right )^{{2}/{3}}+3 \left (9 x^{2}+27\right )^{{1}/{3}}+9\right ) c_{2} \right ) 3^{{2}/{3}}}{9 \left (x^{2}+3\right )^{{2}/{3}}} \]

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 57

ode=3*x^2*(3+x^2)*D[y[x],x]+x*(3+11*x^2)*D[y[x],x]+(1+5*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 \exp \left (\int _1^x-\frac {5 K[1]^2+1}{3 K[1]^4+11 K[1]^3+9 K[1]^2+3 K[1]}dK[1]\right ) \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(x**2 + 3)*Derivative(y(x), (x, 2)) + x*(11*x**2 + 3)*Derivative(y(x), x) + (5*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False