63.15.1 problem 6(a)

Internal problem ID [13184]
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(a)
Date solved : Tuesday, January 28, 2025 at 05:12:04 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }+5 x&=\operatorname {Heaviside}\left (t -2\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 21.588 (sec). Leaf size: 27

dsolve([diff(x(t),t)+5*x(t)=Heaviside(t-2),x(0) = 1],x(t), singsol=all)
 

\[ x \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-5 t +10}}{5}+\frac {\operatorname {Heaviside}\left (t -2\right )}{5}+{\mathrm e}^{-5 t} \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 37

DSolve[{D[x[t],t]+5*x[t]==UnitStep[t-2],{x[0]==1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{-5 t} & t\leq 2 \\ \frac {1}{5} e^{-5 t} \left (5-e^{10}+e^{5 t}\right ) & \text {True} \\ \end {array} \\ \end {array} \]