46.8.6 problem 8

Internal problem ID [7377]
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 : 8
Date solved : Monday, January 27, 2025 at 02:51:18 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=10 \sin \left (t \right )+10 \delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.329 (sec). Leaf size: 47

dsolve([diff(y(t),t$2)+3*diff(y(t),t)+2*y(t)=10*(sin(t)+Dirac(t-1)),y(0) = 1, D(y)(0) = -1],y(t), singsol=all)
 

\[ y = 10 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}-10 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2-2 t}+\sin \left (t \right )-3 \cos \left (t \right )+6 \,{\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \]

✓ Solution by Mathematica

Time used: 0.158 (sec). Leaf size: 46

DSolve[{D[y[t],{t,2}]+3*D[y[t],t]+2*y[t]==10*(Sin[t]+DiracDelta[t-1]),{y[0]==1,Derivative[1][y][0] ==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 10 e^{1-2 t} \left (e^t-e\right ) \theta (t-1)-2 e^{-2 t}+6 e^{-t}+\sin (t)-3 \cos (t) \]