46.8.9 problem 11

Internal problem ID [7380]
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 : 11
Date solved : Monday, January 27, 2025 at 02:51:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=\operatorname {Heaviside}\left (t -1\right )+\delta \left (t -2\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: 0.322 (sec). Leaf size: 68

dsolve([diff(y(t),t$2)+5*diff(y(t),t)+6*y(t)=Heaviside(t-1)+Dirac(t-2),y(0) = 0, D(y)(0) = 1],y(t), singsol=all)
 

\[ y = {\mathrm e}^{-2 t}-{\mathrm e}^{-3 t}-\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-2 t +4}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2-2 t}}{2}+\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{3}+\frac {\operatorname {Heaviside}\left (t -1\right )}{6} \]

✓ Solution by Mathematica

Time used: 0.252 (sec). Leaf size: 80

DSolve[{D[y[t],{t,2}]+5*D[y[t],t]+6*y[t]==UnitStep[t-1]+DiracDelta[t-2],{y[0]==0,Derivative[1][y][0] ==1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{6} e^{-3 t} \left (6 e^4 \left (e^t-e^2\right ) \theta (t-2)-\left (\left (e^t+2 e\right ) \left (e-e^t\right )^2 \theta (1-t)\right )+6 e^t+e^{3 t}-3 e^{t+2}+2 e^3-6\right ) \]