59.1.527 problem 543

Internal problem ID [9699]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 543
Date solved : Sunday, March 30, 2025 at 02:44:39 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 x y&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 32

ode:=x*(x^2+1)*diff(diff(y(x),x),x)+(7*x^2+4)*diff(y(x),x)+8*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\sqrt {x^{2}+1}\, c_2 x +\operatorname {arcsinh}\left (x \right ) c_2 +c_1}{\sqrt {x^{2}+1}\, x^{3}} \]

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 108

ode=x*(1+x^2)*D[y[x],{x,2}]+(4+7*x^2)*D[y[x],x]+8*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {K[1]^2+2}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {7 K[2]^2+4}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {K[1]^2+2}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 1)*Derivative(y(x), (x, 2)) + 8*x*y(x) + (7*x**2 + 4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False