63.15.10 problem 6(j)

Internal problem ID [13114]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 6(j)
Date solved : Wednesday, March 05, 2025 at 09:17:36 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }&=2 x+\operatorname {Heaviside}\left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 8.988 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 25

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

\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {1}{2} \left (-1+e^{2 t-2}\right ) & t>1 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.619 (sec). Leaf size: 26

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

\[ x{\left (t \right )} = \frac {e^{2 t - 2} \theta \left (t - 1\right )}{2} - \frac {\theta \left (t - 1\right )}{2} \]