47.4.7 problem 55

Internal problem ID [7484]
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 : 55
Date solved : Sunday, March 30, 2025 at 12:10:20 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

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

\[ y = c_1 x +c_2 \sqrt {x -1}\, \sqrt {x +1} \]

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \cosh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right )+i c_2 \sinh \left (\frac {\sqrt {1-x^2} \arcsin (x)}{\sqrt {x^2-1}}\right ) \]

✗ Sympy

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

False