68.11.8 problem 20

Internal problem ID [17545]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 20
Date solved : Thursday, October 02, 2025 at 02:25:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {4 \cos \left (t \right )}{3}-\frac {\sin \left (t \right )}{3} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to -\frac {\sin (t)}{3}+\frac {4 \cos (t)}{3}+c_1 \cos (2 t)+c_2 \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.041 (sec). Leaf size: 27

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

\[ y{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )} - \frac {\sin {\left (t \right )}}{3} + \frac {4 \cos {\left (t \right )}}{3} \]