73.22.4 problem 31.6 (d)

Internal problem ID [15631]
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 (d)
Date solved : Tuesday, January 28, 2025 at 08:03:22 AM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=\delta \left (t -1\right )-\delta \left (t -4\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: 9.325 (sec). Leaf size: 23

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

\[ y = \left (-t +4\right ) \operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 87

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

\[ y(t)\to -t \int _1^0(\delta (K[1]-1)-\delta (K[1]-4))dK[1]+\int _1^t\int _1^{K[2]}(\delta (K[1]-1)-\delta (K[1]-4))dK[1]dK[2]-\int _1^0\int _1^{K[2]}(\delta (K[1]-1)-\delta (K[1]-4))dK[1]dK[2] \]