Internal problem ID [10506]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_linear, `class A`]]
\[ \boxed {x^{\prime }+3 x-\left (\delta \left (-1+t \right )\right )-\operatorname {Heaviside}\left (t -4\right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.016 (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)
\[ x \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}+\frac {\operatorname {Heaviside}\left (t -4\right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{3}+{\mathrm e}^{-3 t} \]
✓ Solution by Mathematica
Time used: 0.124 (sec). Leaf size: 50
DSolve[{x'[t]+3*x[t]==DiracDelta[t-1]+UnitStep[t-4],{x[0]==1}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to {cc} \{ & {cc} e^{-3 t} \left (e^3 \theta (t-1)+1\right ) & t\leq 4 \\ \frac {1}{3}-\frac {1}{3} e^{-3 t} \left (-3-3 e^3+e^{12}\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}