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 : Wednesday, March 05, 2025 at 04:24:33 AM
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*}

✓ Maple. Time used: 0.360 (sec). Leaf size: 79

ode:=diff(diff(y(t),t),t)+5*diff(y(t),t)+6*y(t) = Dirac(t-1/2*Pi)+Heaviside(t-Pi)*cos(t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 2.878 (sec). Leaf size: 119

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - pi/2) + 6*y(t) - cos(t)*Heaviside(t - pi) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} + \cos {\left (t \right )} \theta \left (t - \pi \right )\right ) e^{3 t}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} e^{3 t}\, dt + \int \limits ^{0} e^{3 t} \cos {\left (t \right )} \theta \left (t - \pi \right )\, dt\right ) e^{- t} + \int \left (\operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} + \cos {\left (t \right )} \theta \left (t - \pi \right )\right ) e^{2 t}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - \frac {\pi }{2} \right )} e^{2 t}\, dt - \int \limits ^{0} e^{2 t} \cos {\left (t \right )} \theta \left (t - \pi \right )\, dt\right ) e^{- 2 t} \]