14.2.1 problem 1

Internal problem ID [2489]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 1
Date solved : Sunday, March 30, 2025 at 12:03:11 AM
CAS classification : [_separable]

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

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

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

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

✓ Mathematica. Time used: 0.283 (sec). Leaf size: 25

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

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

✓ Sympy. Time used: 0.317 (sec). Leaf size: 8

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

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