59.1.30 problem 31

Internal problem ID [9202]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 31
Date solved : Sunday, March 30, 2025 at 02:25:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

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

\[ y = c_1 \,x^{2}+c_2 \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 115

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

\[ y(x)\to \exp \left (\int _1^x\frac {(K[1]-4) K[1]+2}{2 (K[1]-2) K[1]}dK[1]-\frac {1}{2} \int _1^x\left (-\frac {1}{K[2]}-1+\frac {1}{2-K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]^2-4 K[1]+2}{2 (K[1]-2) K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

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

False