59.1.140 problem 142

Internal problem ID [9312]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 142
Date solved : Wednesday, March 05, 2025 at 07:47:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 27

ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)-3*x*(-x^2+1)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.236 (sec). Leaf size: 107

ode=x^2*(1+x^2)*D[y[x],{x,2}]-3*x*(1-x^2)*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) - 3*x*(1 - x**2)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False