68.12.39 problem 39

Internal problem ID [17633]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 39
Date solved : Thursday, October 02, 2025 at 02:26:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 46

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

\begin{align*} y(t)&\to \frac {1}{4} (\cos (2 t) \arctan (\tan (2 t))+4 c_1 \cos (2 t)+\sin (2 t) (-\log (\cos (2 t))-1+4 c_2)) \end{align*}

✓ Sympy. Time used: 0.231 (sec). Leaf size: 27

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

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