47.4.6 problem 54

Internal problem ID [7483]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number : 54
Date solved : Sunday, March 30, 2025 at 12:10:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 47

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

\[ y = \frac {c_2 \left (x^{2}-1\right ) \ln \left (x -1\right )+\left (-x^{2}+1\right ) c_2 \ln \left (x +1\right )+2 c_1 \,x^{2}-2 c_2 x -2 c_1}{2 x^{2}} \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 56

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

\[ y(x)\to \frac {-4 c_1 x^2-c_2 \left (x^2-1\right ) \log (1-x)+c_2 \left (x^2-1\right ) \log (x+1)+2 c_2 x+4 c_1}{4 x^2} \]

✗ Sympy

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

False