59.1.121 problem 123

Internal problem ID [9293]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 123
Date solved : Sunday, March 30, 2025 at 02:31:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (3+x \right ) y^{\prime }+\left (4+x \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 45

ode:=x^2*(x^2-2*x+1)*diff(diff(y(x),x),x)-x*(x+3)*diff(y(x),x)+(x+4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{2} \left (\operatorname {Ei}_{1}\left (-\frac {4 x}{x -1}\right ) {\mathrm e}^{-\frac {4 x}{x -1}} c_2 +{\mathrm e}^{-\frac {4}{x -1}} c_1 \right )}{x -1} \]

✓ Mathematica. Time used: 0.305 (sec). Leaf size: 116

ode=x^2*(1-2*x+x^2)*D[y[x],{x,2}]-x*(3+x)*D[y[x],x]+(4+x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 - 2*x + 1)*Derivative(y(x), (x, 2)) - x*(x + 3)*Derivative(y(x), x) + (x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False