61.4.15 problem Problem 3(a)

Internal problem ID [15340]
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(a)
Date solved : Thursday, October 02, 2025 at 10:11:46 AM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.145 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 31

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 1 & 0\leq t\leq 2 \\ e^{2-t} & t>2 \\ e^{-t} & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.257 (sec). Leaf size: 39

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

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