58.5.23 problem 23

Internal problem ID [14617]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number : 23
Date solved : Thursday, October 02, 2025 at 09:44:27 AM
CAS classification : [_linear]

\begin{align*} r^{\prime }+r \tan \left (t \right )&=\cos \left (t \right )^{2} \end{align*}

With initial conditions

\begin{align*} r \left (\frac {\pi }{4}\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 16

ode:=diff(r(t),t)+r(t)*tan(t) = cos(t)^2; 
ic:=[r(1/4*Pi) = 1]; 
dsolve([ode,op(ic)],r(t), singsol=all);
 

\[ r = \frac {\left (2 \sin \left (t \right )+\sqrt {2}\right ) \cos \left (t \right )}{2} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 16

ode=D[r[t],t]+r[t]*Tan[t]==Cos[t]^2; 
ic={r[Pi/4]==1}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\begin{align*} r(t)&\to \left (\sin (t)+\frac {1}{\sqrt {2}}\right ) \cos (t) \end{align*}

✓ Sympy. Time used: 0.326 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
r = Function("r") 
ode = Eq(r(t)*tan(t) - cos(t)**2 + Derivative(r(t), t),0) 
ics = {r(pi/4): 1} 
dsolve(ode,func=r(t),ics=ics)
 

\[ r{\left (t \right )} = \left (\sin {\left (t \right )} + \frac {\sqrt {2}}{2}\right ) \cos {\left (t \right )} \]