59.1.528 problem 544

Internal problem ID [9700]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 544
Date solved : Sunday, March 30, 2025 at 02:44:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 31

ode:=2*x^2*(x^2+1)*diff(diff(y(x),x),x)+x*(8*x^2+3)*diff(y(x),x)-(-4*x^2+3)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 x \operatorname {hypergeom}\left (\left [1, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )+\frac {c_2}{x^{{3}/{2}} \left (x^{2}+1\right )^{{1}/{4}}} \]

✓ Mathematica. Time used: 0.175 (sec). Leaf size: 99

ode=2*x^2*(1+x^2)*D[y[x],{x,2}]+x*(3+8*x^2)*D[y[x],x]-(3-4*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + x*(8*x**2 + 3)*Derivative(y(x), x) - (3 - 4*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False