68.12.8 problem 8

Internal problem ID [17602]
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 : 8
Date solved : Thursday, October 02, 2025 at 02:26:08 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\cot \left (4 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

ode:=diff(diff(y(t),t),t)+16*y(t) = cot(4*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (4 t \right ) c_2 +\cos \left (4 t \right ) c_1 +\frac {\sin \left (4 t \right ) \ln \left (\csc \left (4 t \right )-\cot \left (4 t \right )\right )}{16} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 67

ode=D[y[t],{t,2}]+16*y[t]==Cot[4*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.397 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(16*y(t) + Derivative(y(t), (t, 2)) - 1/tan(4*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \cos {\left (4 t \right )} + \left (C_{1} + \frac {\log {\left (\cos {\left (4 t \right )} - 1 \right )}}{32} - \frac {\log {\left (\cos {\left (4 t \right )} + 1 \right )}}{32}\right ) \sin {\left (4 t \right )} \]