67.18.3 problem 27.1 (c)

Internal problem ID [16874]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (c)
Date solved : Thursday, October 02, 2025 at 01:40:01 PM
CAS classification : [[_linear, `class A`]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.097 (sec). Leaf size: 18

ode:=diff(y(t),t)+3*y(t) = Heaviside(t-4); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -4\right ) \left (-1+{\mathrm e}^{-3 t +12}\right )}{3} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 27

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

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{3}-\frac {1}{3} e^{12-3 t} & t>4 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.342 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = - \frac {e^{12 - 3 t} \theta \left (t - 4\right )}{3} + \frac {\theta \left (t - 4\right )}{3} \]