74.14.12 problem 12

Internal problem ID [16346]
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 : 12
Date solved : Monday, March 31, 2025 at 02:50:48 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime }&=\sec \left (2 t \right ) \tan \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

ode:=diff(diff(diff(y(t),t),t),t)+4*diff(y(t),t) = sec(2*t)*tan(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\cos \left (2 t \right ) \left (-\frac {\ln \left (\sec \left (2 t \right )\right )}{2}+\left (t +2 c_1 \right ) \tan \left (2 t \right )+4 \sec \left (2 t \right ) c_3 -2 c_2 +\frac {1}{2}\right )}{4} \]

✓ Mathematica. Time used: 2.15 (sec). Leaf size: 84

ode=D[ y[t],{t,3}]+4*D[y[t],t]==Sec[2*t]*Tan[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{16} \left (\frac {\arctan (\tan (2 t)) (2 t \sin (2 t)+\cos (2 t))}{t}+(4-16 c_2) \cos ^2(t)+2 \cos (2 t) \log (\cos (2 t))+8 c_1 \sin (2 t)+\frac {4 \cos (2 t) \log (\cos (2 t))}{\log \left (\sec ^2(2 t)\right )}\right )+c_3 \]

✓ Sympy. Time used: 0.537 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*Derivative(y(t), t) + Derivative(y(t), (t, 3)) - tan(2*t)/cos(2*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} + \left (C_{2} + \frac {t}{4}\right ) \sin {\left (2 t \right )} + \left (C_{3} + \frac {\log {\left (\cos {\left (2 t \right )} \right )}}{8}\right ) \cos {\left (2 t \right )} \]