73.22.10 problem 31.7 (c)

Internal problem ID [15637]
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 (c)
Date solved : Tuesday, January 28, 2025 at 08:03:27 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }&=\delta \left (t -1\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: 10.255 (sec). Leaf size: 30

dsolve([diff(y(t),t$2)+3*diff(y(t),t)=Dirac(t-1),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{3}+\frac {\operatorname {Heaviside}\left (t -1\right )}{3}-\frac {{\mathrm e}^{-3 t}}{3}+\frac {1}{3} \]

✓ Solution by Mathematica

Time used: 60.043 (sec). Leaf size: 126

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

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