59.1.761 problem 783

Internal problem ID [9933]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 783
Date solved : Sunday, March 30, 2025 at 02:49:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }+x y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 41

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

\[ y(x)\to c_1 x-\frac {c_2 \sqrt [3]{-x^3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )}{3 \sqrt [3]{3}} \]

✗ Sympy

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

False