67.4.33 problem Problem 5(d)

Internal problem ID [14077]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 5(d)
Date solved : Tuesday, January 28, 2025 at 06:13:44 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=1-\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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 9.613 (sec). Leaf size: 41

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

\[ y = -\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}+\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2-2 t}+\frac {{\mathrm e}^{-2 t}}{2}-{\mathrm e}^{-t}+\frac {1}{2} \]

✓ Solution by Mathematica

Time used: 0.070 (sec). Leaf size: 106

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

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