40.8.9 problem 11

Internal problem ID [8666]
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 : Tuesday, September 30, 2025 at 05:40:29 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*}

✓ Maple. Time used: 0.149 (sec). Leaf size: 68

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

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

✓ Mathematica. Time used: 0.335 (sec). Leaf size: 306

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

\begin{align*} y(t)&\to \theta (1-t) \left (e^{-3 t} \left (e^t \int _1^0e^4 \delta (K[2]-2)dK[2]-\int _1^t-e^{3 K[3]} (\delta (K[3]-2)+1)dK[3]-e^t \int _1^te^{2 K[4]} (\delta (K[4]-2)+1)dK[4]+\int _1^0-e^6 \delta (K[1]-2)dK[1]-e^t+1\right )-e^{-3 t} \left (-\int _1^t-e^6 \delta (K[1]-2)dK[1]+e^t \int _1^0e^4 \delta (K[2]-2)dK[2]-e^t \int _1^te^4 \delta (K[2]-2)dK[2]+\int _1^0-e^6 \delta (K[1]-2)dK[1]-e^t+1\right )\right )-e^{-3 t} \left (e^t \int _1^0e^4 \delta (K[2]-2)dK[2]-\int _1^t-e^{3 K[3]} (\delta (K[3]-2)+1)dK[3]-e^t \int _1^te^{2 K[4]} (\delta (K[4]-2)+1)dK[4]+\int _1^0-e^6 \delta (K[1]-2)dK[1]-e^t+1\right ) \end{align*}

✓ Sympy. Time used: 1.072 (sec). Leaf size: 102

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Dirac(t - 2) + 6*y(t) - Heaviside(t - 1) + 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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