46.8.8 problem 10

Internal problem ID [7379]
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 : 10
Date solved : Monday, January 27, 2025 at 02:51:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&=\delta \left (t -\frac {\pi }{2}\right )+\operatorname {Heaviside}\left (t -\pi \right ) \cos \left (t \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: 0.360 (sec). Leaf size: 79

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

\[ y = -\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-3 t +\frac {3 \pi }{2}}+\frac {\operatorname {Heaviside}\left (t -\pi \right ) \left (\cos \left (t \right )+\sin \left (t \right )\right )}{10}-\frac {3 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-3 t +3 \pi }}{10}+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) {\mathrm e}^{-2 t +\pi }+\frac {2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 \pi -2 t}}{5} \]

✓ Solution by Mathematica

Time used: 0.475 (sec). Leaf size: 85

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

\[ y(t)\to \frac {1}{10} e^{-3 t} \left ((\theta (\pi -t)-1) \left (-4 e^{t+2 \pi }-e^{3 t} \sin (t)-e^{3 t} \cos (t)+3 e^{3 \pi }\right )-10 e^{\pi } \left (e^{\pi /2}-e^t\right ) \theta (2 t-\pi )\right ) \]