80.6.24 problem 24

Internal problem ID [21314]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 6. Higher order linear equations. Excercise 6.5 at page 133
Problem number : 24
Date solved : Thursday, October 02, 2025 at 07:28:22 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 67

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

\[ x = -\frac {i \arctan \left ({\mathrm e}^{2 i t}\right )}{4}+c_3 +\frac {{\mathrm e}^{-2 i t} \left (-i \ln \left (\sec \left (2 t \right )\right )+4 i c_1 -4 c_2 -2 t \right )}{16}+\frac {{\mathrm e}^{2 i t} \left (i \ln \left (\sec \left (2 t \right )\right )-4 i c_1 -4 c_2 -2 t \right )}{16} \]

✓ Mathematica. Time used: 0.086 (sec). Leaf size: 63

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

\begin{align*} x(t)&\to \frac {1}{8} \left (-2 t \cos (2 t)-8 c_2 \cos ^2(t)+4 c_1 \sin (2 t)-\log (\cos (t)-\sin (t))+\log (\sin (t)+\cos (t))+\sin (2 t) \log (\cos (2 t))+8 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.345 (sec). Leaf size: 49

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

\[ x{\left (t \right )} = C_{1} + \left (C_{2} - \frac {t}{4}\right ) \cos {\left (2 t \right )} + \left (C_{3} + \frac {\log {\left (\cos {\left (2 t \right )} \right )}}{8}\right ) \sin {\left (2 t \right )} - \frac {\log {\left (\sin {\left (2 t \right )} - 1 \right )}}{16} + \frac {\log {\left (\sin {\left (2 t \right )} + 1 \right )}}{16} \]