74.12.4 problem 4

Internal problem ID [16232]
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 : 4
Date solved : Thursday, March 13, 2025 at 07:59:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=2 \cos \left (4 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)+16*y(t) = 2*cos(4*t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 64

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

\[ y(t)\to \sin (4 t) \int _1^t\frac {1}{2} \cos ^2(4 K[2])dK[2]+\cos (4 t) \int _1^t-\frac {1}{4} \sin (8 K[1])dK[1]+c_1 \cos (4 t)+c_2 \sin (4 t) \]

✓ Sympy. Time used: 0.091 (sec). Leaf size: 19

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

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