67.3.14 problem 4.4 (d)

Internal problem ID [16335]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page 90
Problem number : 4.4 (d)
Date solved : Thursday, October 02, 2025 at 01:20:56 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {1+y^{2}}{x^{2}+1} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 9

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

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

✓ Mathematica. Time used: 0.172 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [\int _1^x\frac {1}{K[2]^2+1}dK[2]+c_1\right ]\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 8

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

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