22.3.4 problem 6.39

Internal problem ID [4517]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 6. The Laplace Transform and Its Applications. Problems at page 291
Problem number : 6.39
Date solved : Tuesday, September 30, 2025 at 07:33:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=8 \sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+4*y(t) = 8*sin(2*t); 
ic:=[y(0) = 1, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 3 \sin \left (2 t \right )+\left (1-2 t \right ) \cos \left (2 t \right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 22

ode=D[y[t],{t,2}]+4*y[t]==8*Sin[2*t]; 
ic={y[0]==1,Derivative[1][y][0] == 4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 3 \sin (2 t)+(1-2 t) \cos (2 t) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - 8*sin(2*t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

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