81.1.43 problem 2-41

Internal problem ID [21488]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 2. Separable differential equations
Problem number : 2-41
Date solved : Thursday, October 02, 2025 at 07:41:48 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 11

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

\[ y = -\tan \left (\arctan \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to -\tan (\arctan (x)-c_1)\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \tan {\left (C_{1} - \operatorname {atan}{\left (x \right )} \right )} \]