28.3.15 problem 6.50

Internal problem ID [4528]
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 : 6.50
Date solved : Monday, January 27, 2025 at 09:22:47 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 3.680 (sec). Leaf size: 33

dsolve([diff(y(t),t$2)+4*y(t)=4*Heaviside(t-Pi)+2*Dirac(t-Pi),y(0) = -1, D(y)(0) = 2],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.192 (sec). Leaf size: 40

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

\[ y(t)\to \theta (t-\pi ) \sin (2 t)-2 \theta (\pi -t) \sin ^2(t)+\sin (2 t)-2 \cos (2 t)+1 \]