68.4.20 problem 20

Internal problem ID [17200]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 20
Date solved : Thursday, October 02, 2025 at 01:51:46 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }&=\frac {\sec \left (t \right )^{2}}{\sec \left (x\right ) \tan \left (x\right )} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 9

ode:=diff(x(t),t) = sec(t)^2/sec(x(t))/tan(x(t)); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \operatorname {arcsec}\left (c_1 +\tan \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.455 (sec). Leaf size: 45

ode=D[x[t],t]==Sec[t]^2/(Sec[x[t]]*Tan[x[t]]); 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\sec ^{-1}(\tan (t)+2 c_1)\\ x(t)&\to \sec ^{-1}(\tan (t)+2 c_1)\\ x(t)&\to -\frac {\pi }{2}\\ x(t)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 0.364 (sec). Leaf size: 20

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

\[ \left [ x{\left (t \right )} = - \operatorname {acos}{\left (C_{1} - \tan {\left (t \right )} \right )} + 2 \pi , \ x{\left (t \right )} = \operatorname {acos}{\left (C_{1} - \tan {\left (t \right )} \right )}\right ] \]