73.22.3 problem 31.6 (c)

Internal problem ID [15630]
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 : 31.6 (c)
Date solved : Tuesday, January 28, 2025 at 08:03:22 AM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\delta \left (t -3\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: 8.797 (sec). Leaf size: 12

dsolve([diff(y(t),t$2)=Dirac(t-3),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \operatorname {Heaviside}\left (t -3\right ) \left (t -3\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 63

DSolve[{D[y[t],{t,2}]==DiracDelta[t-3],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -t \int _1^0\delta (K[1]-3)dK[1]+\int _1^t\int _1^{K[2]}\delta (K[1]-3)dK[1]dK[2]-\int _1^0\int _1^{K[2]}\delta (K[1]-3)dK[1]dK[2] \]