68.14.16 problem 16

Internal problem ID [17708]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 16
Date solved : Thursday, October 02, 2025 at 02:27:15 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime }&=-\sec \left (t \right ) \tan \left (t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

ode:=diff(diff(diff(y(t),t),t),t)+diff(y(t),t) = -sec(t)*tan(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\cos \left (t \right ) \left (-\ln \left (\sec \left (t \right )\right )+\left (-c_1 +t \right ) \tan \left (t \right )-\sec \left (t \right ) c_3 +c_2 +1\right ) \]

✓ Mathematica. Time used: 0.859 (sec). Leaf size: 63

ode=D[ y[t],{t,3}]+D[y[t],t]==-Sec[t]*Tan[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {\arctan (\tan (t)) (t \sin (t)+\cos (t))}{t}-2 (1+c_2) \cos ^2\left (\frac {t}{2}\right )-\cos (t) \log (\cos (t))+c_1 \sin (t)-\frac {2 \cos (t) \log (\cos (t))}{\log \left (\sec ^2(t)\right )}+c_3 \end{align*}

✓ Sympy. Time used: 0.265 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), t) + Derivative(y(t), (t, 3)) + tan(t)/cos(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} + \left (C_{2} - \log {\left (\cos {\left (t \right )} \right )}\right ) \cos {\left (t \right )} + \left (C_{3} - t + \tan {\left (t \right )}\right ) \sin {\left (t \right )} - \frac {1}{\cos {\left (t \right )}} \]