57.5.15 problem 3(c)

Internal problem ID [14370]
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 1.4.1. Integrating factors. Exercises page 41
Problem number : 3(c)
Date solved : Thursday, October 02, 2025 at 09:34:10 AM
CAS classification : [_linear]

\begin{align*} R^{\prime }+\frac {R}{t}&=\frac {2}{t^{2}+1} \end{align*}

With initial conditions

\begin{align*} R \left (1\right )&=3 \ln \left (2\right ) \\ \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 19

ode:=diff(R(t),t)+R(t)/t = 2/(t^2+1); 
ic:=[R(1) = 3*ln(2)]; 
dsolve([ode,op(ic)],R(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 17

ode=D[ R[t],t]+R[t]/t==2/(1+t^2); 
ic={R[1]==Log[8]}; 
DSolve[{ode,ic},R[t],t,IncludeSingularSolutions->True]
 

\begin{align*} R(t)&\to \frac {\log \left (4 t^2+4\right )}{t} \end{align*}

✓ Sympy. Time used: 0.150 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
R = Function("R") 
ode = Eq(Derivative(R(t), t) - 2/(t**2 + 1) + R(t)/t,0) 
ics = {R(1): 3*log(2)} 
dsolve(ode,func=R(t),ics=ics)
 

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