73.22.16 problem 31.7 (i)

Internal problem ID [15643]
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 (i)
Date solved : Tuesday, January 28, 2025 at 08:03:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=\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: 9.622 (sec). Leaf size: 18

dsolve([diff(y(t),t$2)+6*diff(y(t),t)+9*y(t)=Dirac(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
 

\[ y = \left (t -4\right ) {\mathrm e}^{-3 t +12} \operatorname {Heaviside}\left (t -4\right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 86

DSolve[{D[y[t],{t,2}]+6*D[y[t],t]+9*y[t]==DiracDelta[t-4],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to -e^{-3 t} \left (-\int _1^t-4 e^{12} \delta (K[1]-4)dK[1]+t \int _1^0e^{12} \delta (K[2]-4)dK[2]-t \int _1^te^{12} \delta (K[2]-4)dK[2]+\int _1^0-4 e^{12} \delta (K[1]-4)dK[1]\right ) \]