53.2.9 problem 9

Internal problem ID [11302]
Book : Collection of Kovacic problems
Section : section 2. Solution found using all possible Kovacic cases
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 07:37:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{2} \operatorname {hypergeom}\left (\left [\frac {3}{2}, 2\right ], \left [\frac {7}{4}\right ], \frac {x^{2}}{2}\right )+\frac {c_2 \sqrt {x}\, \left (x^{2}+1\right )}{\left (x^{2}-2\right ) \left (-2 x^{2}+4\right )^{{3}/{4}}} \]

✓ Mathematica. Time used: 0.371 (sec). Leaf size: 133

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

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

✗ Sympy

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

False