73.22.13 problem 31.7 (f)

Internal problem ID [15640]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.7 (f)
Date solved : Tuesday, January 28, 2025 at 08:03:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.236 (sec). Leaf size: 5

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

\[ y = 0 \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 37

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

\[ y(t)\to \sin (t) \left (-\int _1^0\delta (K[1])dK[1]\right )+\sin (t) \int _1^t\delta (K[1])dK[1]-\sin (t) \]