85.10.3 problem 2

Internal problem ID [22490]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. B Exercises at page 37
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:41:15 PM
CAS classification : [_separable]

\begin{align*} r^{\prime }&=\frac {\sin \left (t \right )+{\mathrm e}^{r} \sin \left (t \right )}{3 \,{\mathrm e}^{r}+{\mathrm e}^{r} \cos \left (2 t \right )} \end{align*}

With initial conditions

\begin{align*} r \left (\frac {\pi }{2}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.389 (sec). Leaf size: 15

ode:=diff(r(t),t) = (sin(t)+exp(r(t))*sin(t))/(3*exp(r(t))+exp(r(t))*cos(2*t)); 
ic:=[r(1/2*Pi) = 0]; 
dsolve([ode,op(ic)],r(t), singsol=all);
 

\[ r = \ln \left (2 \,{\mathrm e}^{-\frac {\arctan \left (\cos \left (t \right )\right )}{2}}-1\right ) \]

✓ Mathematica. Time used: 1.444 (sec). Leaf size: 19

ode=D[r[t],{t,1}]==  (Sin[t]+Exp[r[t]]*Sin[t]  )/( 3*Exp[r[t]]+Exp[r[t]]*Cos[2*t] ) ; 
ic={r[Pi/2]==0}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\begin{align*} r(t)&\to \log \left (2 e^{-\frac {1}{2} \arctan (\cos (t))}-1\right ) \end{align*}

✓ Sympy. Time used: 2.286 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
r = Function("r") 
ode = Eq(-(exp(r(t))*sin(t) + sin(t))/(exp(r(t))*cos(2*t) + 3*exp(r(t))) + Derivative(r(t), t),0) 
ics = {r(pi/2): 0} 
dsolve(ode,func=r(t),ics=ics)
 

\[ r{\left (t \right )} = \log {\left (2 \left (e^{\int \frac {\sin {\left (t \right )}}{\cos {\left (2 t \right )} + 3}\, dt}\right ) e^{- \int \limits ^{\frac {\pi }{2}} \frac {\sin {\left (t \right )}}{\cos {\left (2 t \right )} + 3}\, dt} - 1 \right )} \]