78.1.31 problem 2 (q)

Internal problem ID [18015]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 1. The Nature of Differential Equations. Separable Equations. Section 2. Problems at page 9
Problem number : 2 (q)
Date solved : Monday, March 31, 2025 at 04:56:41 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 11

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

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

✓ Mathematica. Time used: 0.242 (sec). Leaf size: 29

ode=(1+x^2)*D[y[x],x]+(1+y[x]^2)==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: 0.280 (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)
 

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