59.1.523 problem 539

Internal problem ID [9695]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 539
Date solved : Wednesday, March 05, 2025 at 07:57:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 35

ode:=3*x^2*(-x^2+2)*diff(diff(y(x),x),x)+x*(-11*x^2+1)*diff(y(x),x)+(-5*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} \sqrt {x}}{\left (-2 x^{2}+4\right )^{{3}/{4}}}+c_{2} x^{{1}/{3}} \operatorname {hypergeom}\left (\left [\frac {2}{3}, 1\right ], \left [\frac {11}{12}\right ], \frac {x^{2}}{2}\right ) \]

✓ Mathematica. Time used: 0.286 (sec). Leaf size: 118

ode=3*x^2*(2-x^2)*D[y[x],{x,2}]+x*(1-11*x^2)*D[y[x],x]+(1-5*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {7-5 K[1]^2}{12 K[1]-6 K[1]^3}dK[1]-\frac {1}{2} \int _1^x\frac {1-11 K[2]^2}{6 K[2]-3 K[2]^3}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {7-5 K[1]^2}{12 K[1]-6 K[1]^3}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(2 - x**2)*Derivative(y(x), (x, 2)) + x*(1 - 11*x**2)*Derivative(y(x), x) + (1 - 5*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False