57.4.12 problem 4(d)

Internal problem ID [14336]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number : 4(d)
Date solved : Thursday, October 02, 2025 at 09:31:47 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 13

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

\[ R = \tan \left (\frac {1}{2} t^{2}+c_1 +t \right ) \]

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 31

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

\begin{align*} R(t)&\to \tan \left (\frac {t^2}{2}+t+c_1\right )\\ R(t)&\to -i\\ R(t)&\to i \end{align*}

✓ Sympy. Time used: 0.407 (sec). Leaf size: 12

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

\[ R{\left (t \right )} = \tan {\left (C_{1} + \frac {t^{2}}{2} + t \right )} \]