28.3.4 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 : Monday, January 27, 2025 at 09:22:37 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*}

✓ Solution by Maple

Time used: 2.781 (sec). Leaf size: 21

dsolve([diff(y(t),t$2)+4*y(t)=8*sin(2*t),y(0) = 1, D(y)(0) = 4],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 22

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

\[ y(t)\to 3 \sin (2 t)+(1-2 t) \cos (2 t) \]