40.8.4 problem 6

Internal problem ID [8661]
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 : Tuesday, September 30, 2025 at 05:40:22 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*}

✓ Maple. Time used: 0.276 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = Dirac(t-1); 
ic:=[y(0) = 0, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 105

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

\begin{align*} y(t)&\to -e^{-2 t} \left (\sin (t) \int _1^0e^2 \cos (1) \delta (K[1]-1)dK[1]-\sin (t) \int _1^te^2 \cos (1) \delta (K[1]-1)dK[1]+\cos (t) \int _1^0-e^2 \delta (K[2]-1) \sin (1)dK[2]-\cos (t) \int _1^t-e^2 \delta (K[2]-1) \sin (1)dK[2]-3 \sin (t)\right ) \end{align*}

✓ Sympy. Time used: 1.593 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 1) + 5*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (- \int \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t} \sin {\left (t \right )}\, dt + \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t} \sin {\left (t \right )}\, dt\right ) \cos {\left (t \right )} + \left (\int \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t} \cos {\left (t \right )}\, dt - \int \limits ^{0} \operatorname {Dirac}{\left (t - 1 \right )} e^{2 t} \cos {\left (t \right )}\, dt + 3\right ) \sin {\left (t \right )}\right ) e^{- 2 t} \]