30.11.6 problem 6

Internal problem ID [7586]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 4, Linear Second-Order Equations. EXERCISES 4.1 at page 156
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:54:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 11

ode:=diff(diff(y(t),t),t)+4*y(t) = 2*cos(2*t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\sin \left (2 t \right ) t}{2} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 11

ode=1*D[y[t],{t,2}]+0*D[y[t],t]+4*y[t]==2*Cos[2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t \sin (t) \cos (t) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 10

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

\[ y{\left (t \right )} = \frac {t \sin {\left (2 t \right )}}{2} \]