90.27.11 problem 11

Internal problem ID [25421]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 425
Problem number : 11
Date solved : Friday, October 03, 2025 at 12:01:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=\operatorname {Heaviside}\left (-3+t \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: 0.053 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 37

ode=D[y[t],{t,2}]+2*D[y[t],t]+y[t]==UnitStep[t-3]; 
ic={y[0]==0,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 & t\leq 3 \\ e^{-t} \left (-e^3 (t-2)+e^t+t\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.283 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - Heaviside(t - 3) + 2*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)
 

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