67.4.34 problem Problem 5(e)

Internal problem ID [14078]
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(e)
Date solved : Tuesday, January 28, 2025 at 06:13:45 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }+4 y^{\prime }+y&={\mathrm e}^{-\frac {t}{2}} \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: 9.168 (sec). Leaf size: 17

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

\[ y = \frac {\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) {\mathrm e}^{-\frac {t}{2}}}{4} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 86

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

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