46.8.4 problem 6

Internal problem ID [7375]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.4, page 230
Problem number : 6
Date solved : Monday, January 27, 2025 at 02:51:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=\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 )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.247 (sec). Leaf size: 28

dsolve([diff(y(t),t$2)+4*diff(y(t),t)+5*y(t)=Dirac(t-1),y(0) = 0, D(y)(0) = 3],y(t), singsol=all)
 

\[ y = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2-2 t} \sin \left (t -1\right )+3 \,{\mathrm e}^{-2 t} \sin \left (t \right ) \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 31

DSolve[{D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==DiracDelta[t-1],{y[0]==0,Derivative[1][y][0] ==3}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-2 t} \left (3 \sin (t)-e^2 \theta (t-1) \sin (1-t)\right ) \]