75.20.35 problem 674

Internal problem ID [17098]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 674
Date solved : Monday, March 31, 2025 at 03:41:34 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 }-2 \left (1-x \right ) y&=2 x -2 \end{align*}

With initial conditions

\begin{align*} y \left (\infty \right )&=1 \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 13

ode:=(x^2-2*x)*diff(diff(y(x),x),x)+(-x^2+2)*diff(y(x),x)-2*(1-x)*y(x) = 2*x-2; 
ic:=y(infinity) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\operatorname {signum}\left (c_1 \,x^{2}\right ) \infty \]

✓ Mathematica. Time used: 0.49 (sec). Leaf size: 6

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

\[ y(x)\to 1 \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*Derivative(y(x), (x, 2)) + 2*x*y(x) - 2*x*Derivative(y(x), (x, 2)) - 2*x - 2*y(x) + 2)/(x**2 - 2) cannot be solved by the factorable group method