86.1.3 problem 3

Internal problem ID [23065]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 3. Some standard types of differential equations. Exercise 3b at page 43
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:18:33 PM
CAS classification : [_separable]

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

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

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

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

✓ Mathematica. Time used: 0.172 (sec). Leaf size: 25

ode=1+y[x]^2==(1+x^2)*D[y[x],x]; 
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: 0.202 (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)
 

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