85.71.4 problem 5 (c)

Internal problem ID [22936]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 6. Solution of linear differential equations by Laplace transform. C Exercises at page 284
Problem number : 5 (c)
Date solved : Thursday, October 02, 2025 at 09:16:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=6 \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 )&=0 \\ \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = 6*Dirac(t-2); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 21

ode=D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==6*DiracDelta[t-2]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 6 e^{4-2 t} (t-2) \theta (t-2) \end{align*}

✓ Sympy. Time used: 0.860 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*Dirac(t - 2) + 4*y(t) + 4*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)
 

\[ y{\left (t \right )} = \left (t \left (6 \int \operatorname {Dirac}{\left (t - 2 \right )} e^{2 t}\, dt - 6 \int \limits ^{0} \operatorname {Dirac}{\left (t - 2 \right )} e^{2 t}\, dt\right ) - 6 \int t \operatorname {Dirac}{\left (t - 2 \right )} e^{2 t}\, dt + 6 \int \limits ^{0} t \operatorname {Dirac}{\left (t - 2 \right )} e^{2 t}\, dt\right ) e^{- 2 t} \]