68.11.9 problem 21

Internal problem ID [17546]
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 : 21
Date solved : Thursday, October 02, 2025 at 02:25:17 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.432 (sec). Leaf size: 78

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

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

✓ Sympy. Time used: 0.055 (sec). Leaf size: 22

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

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