59.1.20 problem 20

Internal problem ID [9192]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 20
Date solved : Sunday, March 30, 2025 at 02:25:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

ode:=(x^2+1)*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,x^{3}-3 c_1 \,x^{2}-3 c_2 x +c_1 \]

✓ Mathematica. Time used: 0.267 (sec). Leaf size: 75

ode=(1+x^2)*D[y[x],{x,2}]-4*x*D[y[x],x]+6*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False