76.21.15 problem 14 (d)

Internal problem ID [17779]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.7 (Impulse Functions). Problems at page 350
Problem number : 14 (d)
Date solved : Tuesday, January 28, 2025 at 11:02:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 11.816 (sec). Leaf size: 13

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

\[ y = \operatorname {Heaviside}\left (t -1\right ) \sin \left (t -1\right ) \]

✓ Solution by Mathematica

Time used: 0.065 (sec). Leaf size: 17

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

\[ y(t)\to -\theta (t-1) \sin (1-t) \]