63.17.1 problem 2

Internal problem ID [13200]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page 173
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 05:12:19 AM
CAS classification : [[_linear, `class A`]]

\begin{align*} x^{\prime }+3 x&=\delta \left (t -1\right )+\operatorname {Heaviside}\left (t -4\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.410 (sec). Leaf size: 38

dsolve([diff(x(t),t)+3*x(t)=Dirac(t-1)+Heaviside(t-4),x(0) = 1],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.684 (sec). Leaf size: 132

DSolve[{D[x[t],t]+3*x[t]==DiracDelta[t-1]+UnitStep[t-4],{x[0]==1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \theta (4-t) \left (\frac {1}{3} e^{-3 t} \left (-3 \int _1^te^{3 K[2]} (\delta (K[2]-1)+1)dK[2]+e^3 (3 \theta (0)-4)+e^{12}-3\right )+e^{-3 t} \left (\int _0^te^3 \delta (K[1]-1)dK[1]+1\right )\right )-\frac {1}{3} e^{-3 t} \left (-3 \int _1^te^{3 K[2]} (\delta (K[2]-1)+1)dK[2]+e^3 (3 \theta (0)-4)+e^{12}-3\right ) \]