20.23.9 problem Problem 9

Internal problem ID [3981]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 10, The Laplace Transform and Some Elementary Applications. Exercises for 10.8. page 710
Problem number : Problem 9
Date solved : Tuesday, March 04, 2025 at 05:22:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 y&=\delta \left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 5.524 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+3*y(t) = Dirac(t-2); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.963 (sec). Leaf size: 63

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

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