59.1.563 problem 579

Internal problem ID [9735]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 579
Date solved : Sunday, March 30, 2025 at 02:45:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.176 (sec). Leaf size: 81

ode:=2*x^2*(x^2+2)*diff(diff(y(x),x),x)+7*x^3*diff(y(x),x)+(3*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sqrt {x}\, \left (2^{{3}/{4}} c_1 +\ln \left (-\sqrt {2}\, \left (2 x^{2}+4\right )^{{1}/{4}}+2\right ) c_2 -\ln \left (\sqrt {2}\, \left (2 x^{2}+4\right )^{{1}/{4}}+2\right ) c_2 +2 \arctan \left (\frac {\sqrt {2}\, \left (2 x^{2}+4\right )^{{1}/{4}}}{2}\right ) c_2 \right ) 2^{{1}/{4}}}{2 \left (x^{2}+2\right )^{{3}/{4}}} \]

✓ Mathematica. Time used: 0.429 (sec). Leaf size: 93

ode=2*x^2*(2+x^2)*D[y[x],{x,2}]+7*x^3*D[y[x],x]+(1+3*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {3 K[1]^2+4}{4 K[1]^3+8 K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]^2+4}{4 K[1]^3+8 K[1]}dK[1]\right )dK[2]+c_1\right )}{\left (x^2+2\right )^{7/8}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x**3*Derivative(y(x), x) + 2*x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + (3*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False