59.1.48 problem 50

Internal problem ID [9220]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 50
Date solved : Sunday, March 30, 2025 at 02:25:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 42

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

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

✓ Mathematica. Time used: 0.788 (sec). Leaf size: 73

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

\[ y(x)\to \frac {\left (c_1 x-c_2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},-2 x^2\right )\right ) \exp \left (-\frac {1}{2} \int _1^x\frac {2 \left (K[1]^2+2\right )}{2 K[1]^3+K[1]}dK[1]\right )}{\left (2 x^2+1\right )^{3/4}} \]

✗ Sympy

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

False