47.4.3 problem 51

Internal problem ID [7480]
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 : 51
Date solved : Sunday, March 30, 2025 at 12:10:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

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

\[ y = c_1 \sin \left (\operatorname {arcsinh}\left (x \right )\right )+c_2 \cos \left (\operatorname {arcsinh}\left (x \right )\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \cos (\text {arcsinh}(x))+c_2 \sin (\text {arcsinh}(x)) \]

✗ Sympy

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

False