68.12.36 problem 36

Internal problem ID [17630]
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 : 36
Date solved : Thursday, October 02, 2025 at 02:26:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\tan \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 56

ode:=diff(diff(y(t),t),t)+16*y(t) = tan(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\ln \left (\cos \left (2 t \right )\right ) \cos \left (2 t \right ) \sin \left (2 t \right )}{4}+\frac {\left (-t +4 c_1 \right ) \cos \left (2 t \right )^{2}}{2}+\frac {\sin \left (2 t \right ) \left (16 c_2 -1\right ) \cos \left (2 t \right )}{8}+\frac {t}{4}-c_1 \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 61

ode=D[y[t],{t,2}]+16*y[t]==Tan[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos (4 t) \int _1^t-\frac {1}{2} \sin ^2(2 K[1])dK[1]+c_1 \cos (4 t)-\frac {1}{16} \sin (4 t) (\cos (4 t)-2 \log (\cos (2 t))+1-16 c_2) \end{align*}

✓ Sympy. Time used: 9.522 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*y(t) - tan(2*t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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