59.1.109 problem 111

Internal problem ID [9281]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 111
Date solved : Sunday, March 30, 2025 at 02:31:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 35

ode:=9*x^2*diff(diff(y(x),x),x)+3*x*(x^2+3)*diff(y(x),x)-(-5*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \,{\mathrm e}^{-\frac {x^{2}}{12}} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {1}{6}, \frac {x^{2}}{6}\right )}{x}+\frac {c_2 \,{\mathrm e}^{-\frac {x^{2}}{6}}}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.456 (sec). Leaf size: 70

ode=9*x^2*D[y[x],{x,2}]+3*x*(3+x^2)*D[y[x],x]-(1-5*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\frac {x^2}{6}-\frac {1}{6}} \left (2 \sqrt [3]{e} c_1 x^{4/3}+\sqrt [3]{6} c_2 \left (-x^2\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^2}{6}\right )\right )}{2 x^{5/3}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + 3*x*(x**2 + 3)*Derivative(y(x), x) - (1 - 5*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False