61.4.18 problem Problem 3(d)

Internal problem ID [15343]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 3(d)
Date solved : Thursday, October 02, 2025 at 10:11:48 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=\operatorname {Heaviside}\left (t \right )-\operatorname {Heaviside}\left (t -1\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: 0.151 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 43

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-t} & t<0 \\ 1-e^{-t} t & 0\leq t\leq 1 \\ (-1+e) e^{-t} t & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.446 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - Heaviside(t) + Heaviside(t - 1) + 2*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)
 

\[ y{\left (t \right )} = \left (t \left (- \theta \left (t\right ) + e \theta \left (t - 1\right )\right ) - \theta \left (t\right ) + 1\right ) e^{- t} + \theta \left (t\right ) - \theta \left (t - 1\right ) \]