59.1.264 problem 267

Internal problem ID [9436]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 267
Date solved : Sunday, March 30, 2025 at 02:34:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

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

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

✓ Mathematica. Time used: 0.319 (sec). Leaf size: 43

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

\[ y(x)\to e^{-x} (x-3) x \left (c_2 \int _1^x\frac {e^{K[1]}}{(K[1]-3)^2 K[1]^3}dK[1]+c_1\right ) \]

✗ Sympy

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

False