59.1.191 problem 193

Internal problem ID [9363]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 193
Date solved : Wednesday, March 05, 2025 at 07:48:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 27

ode:=4*x^2*(x^2+1)*diff(diff(y(x),x),x)+4*x*(x^2+2)*diff(y(x),x)-(x^2+15)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{2} \left (x^{2}+1\right )^{{3}/{2}}+3 c_{1} x^{2}+2 c_{1}}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 110

ode=4*x^2*(1+x^2)*D[y[x],{x,2}]+4*x*(2+x^2)*D[y[x],x]-(15+x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 4*x*(x**2 + 2)*Derivative(y(x), x) - (x**2 + 15)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False