53.1.292 problem 295

Internal problem ID [10764]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 295
Date solved : Tuesday, September 30, 2025 at 07:31:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 39

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]
 

\begin{align*} y(x)&\to c_2 x \text {arctanh}\left (\frac {x}{\sqrt {x^2+1}}\right )-c_2 \sqrt {x^2+1}+c_1 x \end{align*}

✗ 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