34.2.7 problem 7

Internal problem ID [6037]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 2, Equations of the first order and degree. page 20
Problem number : 7
Date solved : Wednesday, March 05, 2025 at 12:10:20 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (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: 0.274 (sec). Leaf size: 25

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

✓ Sympy. Time used: 0.268 (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)
 

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