63.4.12 problem 4(d)

Internal problem ID [12976]
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 : Wednesday, March 05, 2025 at 08:55:32 PM
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);
 

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

✓ Mathematica. Time used: 0.238 (sec). Leaf size: 48

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

✓ Sympy. Time used: 0.616 (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)
 

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