68.12.7 problem 7

Internal problem ID [17601]
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 : 7
Date solved : Thursday, October 02, 2025 at 02:26:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=\csc \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) = csc(4*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 37

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

\begin{align*} y(t)&\to \left (-\frac {t}{4}+c_1\right ) \cos (4 t)+\frac {1}{16} \sin (4 t) (\log (\sin (4 t))+16 c_2) \end{align*}

✓ Sympy. Time used: 0.257 (sec). Leaf size: 27

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

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