90.29.1 problem 9

Internal problem ID [25433]
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 449
Problem number : 9
Date solved : Friday, October 03, 2025 at 12:01:30 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-3 y&=\operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 29

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

\[ y = -\frac {\operatorname {Heaviside}\left (t -2\right )}{3}+2 \,{\mathrm e}^{3 t}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{3 t -6}}{3} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 38

ode=D[y[t],t]-3*y[t]==UnitStep[t-2]; 
ic={y[0]==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 2 e^{3 t} & t\leq 2 \\ \frac {1}{3} \left (-1+6 e^{3 t}+e^{3 t-6}\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.402 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*y(t) - Heaviside(t - 2) + Derivative(y(t), t),0) 
ics = {y(0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 e^{3 t} + \frac {e^{3 t - 6} \theta \left (t - 2\right )}{3} - \frac {\theta \left (t - 2\right )}{3} \]