28.3.11 problem 6.46

Internal problem ID [4524]
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 : 6.46
Date solved : Monday, January 27, 2025 at 09:22:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 3.299 (sec). Leaf size: 33

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

\[ y = \left (\left (-2 t +2 \pi \right ) \cos \left (2 t \right )+\sin \left (2 t \right )\right ) \operatorname {Heaviside}\left (t -\pi \right )+\sin \left (2 t \right ) \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 32

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \sin (2 t) & t\leq \pi \\ 2 ((\pi -t) \cos (2 t)+\sin (2 t)) & \text {True} \\ \end {array} \\ \end {array} \]