73.22.12 problem 31.7 (e)

Internal problem ID [15639]
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 (e)
Date solved : Tuesday, January 28, 2025 at 08:03:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-16 y&=\delta \left (t -10\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: 10.603 (sec). Leaf size: 16

dsolve([diff(y(t),t$2)-16*y(t)=Dirac(t-10),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\operatorname {Heaviside}\left (t -10\right ) \sinh \left (-40+4 t \right )}{4} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 104

DSolve[{D[y[t],{t,2}]-16*y[t]==DiracDelta[t-10],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^{-4 t} \left (e^{8 t} \int _1^0\frac {\delta (K[1]-10)}{8 e^{40}}dK[1]-e^{8 t} \int _1^t\frac {\delta (K[1]-10)}{8 e^{40}}dK[1]-\int _1^t-\frac {1}{8} e^{40} \delta (K[2]-10)dK[2]+\int _1^0-\frac {1}{8} e^{40} \delta (K[2]-10)dK[2]\right ) \]